\section{Introduction}
\sloppy

Since the introduction by \cite{cox:1981}, \acrfullpl{odm} have been receiving renewed interest in recent years. These models are widely applied in various fields ranging from economics (see \cite{pindyck1998econometric}), environmental study (see \cite{bhaskaran2013time}), epidemiology and public health study (see \cite{zeger1988regression, davis1999modeling, ferland:latour:oraichi:2006}), finance (see \cite{liesenfeld2003univariate, rydberg2003dynamics,fokianos:tjostheim:2011,francq2011garch}) and population dynamics (see \cite{ives2003estimating}). The celebrated \acrshort{garch}$(1,1)$ model introduced in
\cite{bollerslev:1986} as well as most of the models derived from this one are typical
examples of \acrshortpl{odm}; see \cite{bollerslev08-glossary} for a list of some of them. A list of contributions on this class of models specifically dealing with discrete data includes
\cite{streett:2000, davis:dunsmuir:streett:2003, heinen2003modelling, ferland:latour:oraichi:2006, fokianos:rahbek:tjostheim:2009, franke2010weak, fokianos:tjostheim:2011, henderson:matteson:woodard:2011, neuman:2011, davis:liu:2012, doukhan:fokianos:tjostheim:2012, dou:kou:mou:2013, fokianos2013goodness, christou2014quasi, christou2015count} and \cite{douc2015handy}.

\acrshortpl{odm} have the nice
feature that the computations of the associated (conditional) likelihood and its derivatives are
easy, the parameter estimation is hence relatively simple, and the prediction, which is a prime objective in many time series applications, is straightforward. However, it turns out that the asymptotic properties of the \acrfull{mle} for this class can
be cumbersome to establish, except when they can be derived using computations specific to the
studied model (the \acrshort{garch}$(1,1)$ case being one of the most celebrated
example). The literature concerning the asymptotic theory of the \acrshort{mle} when the observed variable has Poisson distribution includes \cite{fokianos:rahbek:tjostheim:2009, fokianos:tjostheim:2011, fokianos2012nonlinear} and \cite{wang2014self}. For a more general case where the model belongs to the class of one-parameter exponential \acrshortpl{odm}, such as the Bernoulli, the exponential, the negative binomial (with known frequency parameter) and the Poisson autoregressive models, the consistency and the asymptotic normality of the \acrshort{mle} have been derived in \cite{davis:liu:2012}. However, the one-parameter exponential family is inadequate to deal with  models such as multi-parametric, mixture or multivariate \acrshortpl{odm} (the negative binomial with all unknown parameters and mixture Poisson \acrshortpl{odm} are examples of this case). A more general consistency result, yet not the asymptotic normality, has been obtained recently in \cite{dou:kou:mou:2013}. This general result allows the observed process to admit various forms of distribution and to take values in any Borel space, and allows the hidden process to assume values in any locally compact Polish space endowed with the associated Borel $\sigma$-field. This result has later been extended and refined in \cite{douc2015handy}. However, most of the results obtained so far have been derived only under the framework of \acrshort{garch}$(1,1)$-type or first-order \acrshortpl{odm}, yet less is known for the \acrshort{garch}$(p,q)$-type discrete \acrshortpl{odm}, as highlighted as a remaining unsolved problem in \cite{tjostheim2015count}. 


In this contribution, we consider among others (see \cite{streett:2000, heinen2003modelling}) a general class of \acrshortpl{odm} that is capable to account for several lagged variables of both hidden and observation processes. Namely, we develop theory and inference for the class of general-order \acrshortpl{odm} parallel to the \acrshort{garch}$(p,q)$ family, under the assumption of well-specified models. For the development on the \acrshort{garch}$(p,q)$ model, see for example \cite{francq2004maximum, francq2011garch} and for multivariate case, see \cite{comte2003asymptotic}.
We extend the approaches introduced in \cite{dou:kou:mou:2013} and later used in \cite{douc2015handy} to establish the consistency of the \acrshort{mle} and the ergodicity for the models in this general context. These results can in principle be obtained by embedding the studied model into the corresponding first-order one and then applying the results obtained therein to the embedded model. By appropriate adjustments,
the consistency of the \acrshort{mle} for the general-order \acrshort{odm} can be derived. Yet for ergodicity, the generalized result seems not trivial; it turns out that more general assumptions are required, compared to the usual first-order \acrshortpl{odm}. In either \cite{dou:kou:mou:2013} or \cite{douc2015handy}, the ergodicity of the usual first-order \acrshortpl{odm} is obtained by showing the uniqueness of invariant probability measure for the hidden process, which is by its own right a Markov chain. In these papers, showing the existence of the invariant probability measure for the hidden process relies on the Foster-Lyapunov-type assumption; however, for general \acrshortpl{odm} or embedded ones, this assumption may be replaced by the \emph{iterative} Foster-Lyapunov-type assumption instead (see Assumption~\ref{assum:weakFeller-V:gen} of \autoref{sec:ergodicity:gen:od}).
The asymptotic normality of the \acrshort{mle} is also investigated in this general setting.
In this study, we restrict our consideration to the special yet important case where the current hidden state variable is expressed as a linear function with respect to its own past variables. This class is rich enough to cover most of the familiar \acrshortpl{odm}, such as the \acrshort{garch}$(p,q)$, the log-linear Poisson autoregression of order $(p,q)$ and the \acrshort{nbingarch}$(p,q)$ models, and other instances such as multi-parametric, mixture or multivariate \acrshortpl{odm}. To establish the asymptotic normality, we follow the classical approach by first approximating the score function by the stationary version of it and then developing the stationary score function around the true parameter through a Taylor expansion. By appropriate assumptions, the central limit theorem for Martingale difference applies and the stationary score function can then be shown to be asymptotically Gaussian. Then assuming invertibility of the asymptotic Fisher information matrix, the asymptotic normality follows. All the results are presented under sufficient and easy-to-check conditions. As a demonstration, they are then applied to the log-linear Poisson autoregression of order $(p,q)$ and the \acrshort{nbingarch}$(p,q)$ models. Finally, we provide an empirical study suggesting that in some circumstance higher-order model may fit the data better than the first-order one if the class of \acrshortpl{odm} is used.

This {paper} is structured as follows. Specific definitions and notation are
introduced in \autoref{sec:definitions-notation-od-gen}. ~\autoref{sec:main-result:gen:od} presents the main results of consistency and asymptotic normality of the \gls{mle} as well as ergodic solution for the model. We apply these results to some specific models in \autoref{sec:examples:gen:od}.
Numerical experiments is given in \autoref{sec:empir:study:gen:od}. Finally, \autoref{sec:append:proofs:gen:od} contains the postponed proofs.

\section{Definitions and Notation}
\label{sec:definitions-notation-od-gen}
Let $(\Xset,\Xmet)$ be a complete and separable metric space
endowed with the associated Borel $\sigma$-field $\Xsigma$ and
$(\Yset,\Ysigma)$ be a Borel space. Let $(\Theta,\met)$, the set of parameters,
be a compact metric space equipped with the metric $\met$, $\{G^\theta\,:\,\theta\in\Theta\}$ be a family of probability kernels on $\Xset \times \Ysigma$ and $p, q$ be two positive integers. The observation-driven time series model can be formally defined as
follows.

%Consider a bivariate stochastic process $\{(X_k,Y_k)\,:\,k\in\zsetp\}$ on
%$\Xset\times\Yset$, where $(\Xset,\Xmet)$ is a complete and separable metric space
%endowed with the associated Borel $\sigma$-field $\Xsigma$ and
%$(\Yset,\Ysigma)$ is a Borel space. Let $(\Theta,\met)$, the set of parameters,
%be a compact metric space, $\{G^\theta\,:\,\theta\in\Theta\}$ be a family of
%probability kernels on $\Xset \times \Ysigma$ and $p, q$ are two positive integers. The observation-driven time series model can be formally defined as
%follows.



\begin{definition}\label{def:obs-driv-gen}
  A time series $\nsequence{Y_k}{k \ge -q+1}$ %$\{Y_k \,:\,k\in\zsetp\}$
  valued in $\Yset$ is said to be
  distributed according to an \emph{\acrlong{odm} of order $(p,q)$} (hereafter, \acrshort{odm}$(p, q)$) with parameter
  $\theta\in\Theta$ if there exist a family of measurable
functions $\{(\chunk{x}{1}{p},\chunk{y}{1}{q})
\mapsto \psi^\theta_{\chunk{y}{1}{q}}(\chunk{x}{1}{p})\,:\, \theta \in \Theta\}$ from $(\Xset^p\times \Yset^q, \Xsigma^{\otimes p} \otimes \Ysigma^{\otimes q})$ to $(\Xset,
\Xsigma)$ and a process $\nsequence{X_k}{k \ge -p+1}$ on $(\Xset,
\Xsigma)$ such that
%the process $\{(X_k,Y_k)\,:\,k\in\zsetp\}$ on $\Xset\times\Yset$ satisfies
for all $k\in\zsetp$,
\begin{equation}\label{eq:def:gen-ob}
\begin{split}
&X_{k+1}=\psi^\theta_{\chunk{Y}{k-q+1}{k}}(\chunk{X}{k-p+1}{k})\eqsp, \\
&Y_{k+1}\mid \mathcal{F}_k\sim G^\theta(X_{k+1};\cdot)\eqsp,
\end{split}
\end{equation}
where $\mathcal{F}_k=\sigma\left(\chunk{X}{-p+1}{k+1},\chunk{Y}{-q+1}{k}\right)$ and $\chunk{u}{\ell}{m}\eqdef(u_\ell,\ldots,u_m)$ for $\ell \leq m$. Moreover, we say
that the model is dominated by  some $\sigma$-finite measure $\nu$ on
$(\Yset,\Ysigma)$ if for all $x \in \Xset$, the probability kernel $G^\theta (x;\cdot)$ is dominated by
$\nu$. In this case we denote by $g^\theta (x;\cdot)$ its Radon--Nikodym
derivative,
$g^\theta (x;y)=\frac{\rmd G^\theta (x;\cdot)}{\rmd
  \nu}(y)$, and we always assume that for all $(x,y) \in \Xset \times \Yset$ and for all $\theta\in\Theta$,
\begin{equation*}
g^\theta(x;y)>0\;.
\end{equation*}
\end{definition}

One of the most popular examples of this class is the general \acrshort{garch}$(p,q)$ model introduced by \cite{bollerslev:1986}, where $\Xset=(0,\infty)$, $\Yset=\rset$, $G^\theta(x;\cdot)$ is the centered Gaussian distribution of variance $x$, the deterministic function $\psi^\theta_{\chunk{y}{1}{q}}(\chunk{x}{1}{p})=\omega+\sum_{i=1}^p a_i x_i +\sum_{i=1}^q b_i y_i^2$ and $\theta=(\omega,\chunk{a}{1}{p},\chunk{b}{1}{q})$ with $\omega>0$ and $\chunk{a}{1}{p}, \chunk{b}{1}{q}\ge0$. This model was then extensively studied by, for example, \cite{bougerol:picard:1992, francq2004maximum, francq2009tour, lindner2009stationarity, francq2011garch} and the references therein. For other \acrshort{garch} examples of this class, see \cite{bollerslev08-glossary}.
\begin{remark}
When $p=q=1$, then the \acrshort{odm}$(p,q)$ defined by \eqref{eq:def:gen-ob} collapses to the first-order \acrshort{odm} considered in \cite{dou:kou:mou:2013} and \cite{douc2015handy}.
Note also that setting $r\eqdef\max(p,q)$ and defining $\varphi^\theta:\Xset^r\times\Yset^r\to\Xset$ by, for all $(\chunk{x}{-r+1}{0},\chunk{y}{-r+1}{0})\in\Xset^r\times\Yset^r$,
\begin{equation*}
\varphi^\theta_{\chunk{y}{-r+1}{0}}(\chunk{x}{-r+1}{0})\eqdef\psi^\theta_{\chunk{y}{-q+1}{0}}(\chunk{x}{-p+1}{0})\eqsp,
\end{equation*}
then the \acrshort{odm}($p,q$) can be generally embedded in an \acrshort{odm}($r,r$). Thus without loss of generality, we can always assume that  $p=q$.
\end{remark}

The inference about the model parameter is performed by relying on the conditional
likelihood of the observations $(Y_1,\ldots,Y_n)$ given $(\chunk{X}{-p+1}{0},\chunk{Y}{-q+1}{0})=(\chunk{x}{-p+1}{0},\chunk{y}{-q+1}{0})\eqdef z$ for an arbitrary
$z\in\Xset^p\times \Yset^q$. The corresponding conditional density function with respect to $\nu^{\otimes
  n}$ is, under parameter $\theta\in\Theta$, for all $z=(\chunk{x}{-p+1}{0},\chunk{y}{-q+1}{0})\in\Xset^p\times \Yset^q$,
\begin{equation} \label{eq:lkd:Y:cl:condX1:gen}
\chunk{y}{1}{n} \mapsto \prod_{k=0}^{n-1} g^\theta\left(\f{\chunk{y}{-q+1}{k}}(\chunk{x}{-p+1}{0});y_{k+1}\right) \eqsp,
\end{equation}
where, for any vector $\chunk{y}{0}{k}$,
the function $\f{\chunk{y}{0}{k}}:\Xset^p\times \Yset^q\to\Xset$ is successively defined by, for all $k\in\zsetp$,
\begin{multline}
\f{\chunk{y}{-q+1}{k}}(\chunk{x}{-p+1}{0})  \\
\eqdef\psi_{\chunk{y}{k-q+1}{k}}^\theta\left(\f{\chunk{y}{-q+1}{k-p}}(\chunk{x}{-p+1}{0}),\ldots, \f{\chunk{y}{-q+1}{k-1}}(\chunk{x}{-p+1}{0})\right)
\end{multline}
with the convention, for $1\le j\le p$,
\begin{equation*}
\f{\chunk{y}{-q+1}{-j}}(\chunk{x}{-p+1}{0})\eqdef x_{-j+1} \eqsp.
\end{equation*}
%\rd{In the case when $-q+1>-j$ (which can happen if $q<p$), do we still use the convention $\f{\chunk{y}{-q+1}{-j}}(\chunk{x}{-p+1}{0})\eqdef x_{-j+1}$? }
Then, the corresponding
(conditional) \acrfull{mle} $\mlY{z,n}$ of the parameter $\theta$, with $z=(\chunk{x}{-p+1}{0},\chunk{y}{-q+1}{0})$,
 is defined by
\begin{equation}
\label{eq:defi:mle-od:gen}
\mlY{z,n} \in \argmax_{\theta \in \Theta} \mathsf{L}_{z,n}^\theta\eqsp,
\end{equation}
where
\begin{equation}
\label{eq:defi:lkdM:gen:od}
\mathsf{L}_{z,n}^\theta\eqdef   n^{-1} \sum_{k=0}^{n-1} \ln g^\theta\left(\f{\chunk{y}{-q+1}{0},\chunk{Y}{1}{k}}(\chunk{x}{-p+1}{0});Y_{k+1}\right) \eqsp.
\end{equation}
In this contribution, we investigate the convergence of $\mlY{z,n}$ as $n\to\infty$
for some (well-chosen) value of $z$ under the assumption that the model is well specified and
the observations are in a steady state. That is, we assume that the
observations $\{Y_k \,:\,k\in\zsetp\}$ are distributed according to
$\tilde\PP^{\thv}$ with some $\thv\in\Theta$, where, for all $\theta\in\Theta$,
$\tilde\PP^{\theta}$ denotes the stationary distribution of the observation-driven time series corresponding to the parameter $\theta$. However, whether
such a distribution is well defined is not always obvious. Let us now detail how this probability distribution  can be obtained.

For simplicity, we assume that $p=q$ and that $p\ge 2$ and let $\Zset=\Xset^p\times\Yset^{p-1}$ and $\Zsigma=\Xsigma^{\otimes p}\otimes\Ysigma^{\otimes {(p-1)}}$.
Let $\Psi^\theta\,:\,\Zset\times\Yset\to\Zset$ be defined by, for all $z=(\chunk{z}{1}{2p-1})\in\Zset$ and $y\in\Yset$, %\mo{Shall we set $f(z_{p+2}),\ldots,f(y)$ instead of $z_{p+2},\ldots, y$?}
\begin{align}\label{eq:def:psi:gen:od}
\begin{split}
\Psi^\theta_{y}(z)\eqdef\left(z_{2},\ldots,z_{p}, \psi^\theta_{\chunk{z}{p+1}{2p-1}, {y}}(\chunk{z}{1}{p}), z_{p+2},\ldots, z_{2p-1}, y\right)
\end{split}
\end{align}

%\begin{align}\label{eq:def:psi:gen:od}
%\begin{split}
%\Psi^\theta_{y}(z)\eqdef\left(\begin{array}{c}z_{2} \\\vdots \\z_{p} \\ \psi^\theta_{\chunk{z}{p+1}{2p-1},{y}}(\chunk{z}{1}{p})\\z_{p+2} \\\vdots \\z_{2p-1}\\y\end{array}\right)
%\end{split}
%\end{align}

The function $\Psi^\theta$ is indeed measurable on $(\Zset\times\Yset,\Zsigma\otimes\Ysigma)$. Moreover, for any vector $\chunk{y}{1}{k}\in\Yset^k$,
we define $\F{\chunk{y}{1}{k}}$ as a $\Zset\to\Zset$ function obtained by the successive
composition of  $\Psi^\theta_{y_1}$,  $\Psi^\theta_{y_2}$, ..., and $\Psi^\theta_{y_k}$,
\begin{equation}
\label{eq:notationItere:f:cl:gen}
\F{\chunk{y}{1}{k}}=\Psi^\theta_{y_k} \circ \Psi^\theta_{y_{k-1}} \circ \dots \circ \Psi^\theta_{y_1}\,.
\end{equation}
with the convention $\F{\chunk{y}{1}{0}}(z)=z$.
By letting $Z_k=(\chunk{X}{k-p+1}{k},\chunk{Y}{k-p+1}{k-1})$ and observing that $\mathcal{F}_k=\sigma\left(\chunk{Z}{0}{k+1},\chunk{Y}{0}{k}\right)$, Model~\eqref{eq:def:gen-ob} can be replaced by: for all $k\in\zsetp$,

\begin{equation}\label{eq:def:ob:standard}
\begin{split}
&Z_{k+1}=\Psi^\theta_{Y_k}(Z_k)\eqsp, \\
&Y_{k+1}\mid \mathcal{F}^{Z,Y}_k\sim H^\theta(Z_{k+1};\cdot)\eqsp,
\end{split}
\end{equation}
where, for all $z=(\chunk{z}{1}{2p-1})\in\Zset$,
\begin{equation}\label{eq:def:H:G:gen}
H^\theta(z;\cdot)\eqdef G^\theta(\Proj{p}{z};\cdot)\eqsp
\end{equation}
and, for all $j\in\{1,\ldots,2p-1\}$, $\Proj{j}{z}=z_j$.
By this representation, the \acrshort{odm}$(p,p)$ is thus imbedded in an \acrshort{odm}$(1,1)$. This in principle allows us to apply the same results obtained for the class of \acrshortpl{odm}$(1,1)$ to the broader class of \acrshortpl{odm}$(p,p)$. As an \acrshort{odm}$(1,1)$, the bivariate process $\dsequence{Z}{Y}[k][\zsetp]$  is a Markov chain on the space $(\Zset\times\Yset,\Zsigma\otimes\Ysigma)$ with transition kernel $K^\theta$ satisfying, for all $(z,y)\in\Zset\times\Yset$, $A\in\Zsigma$ and $B\in\Ysigma$,
\begin{equation}\label{eq:def:kernel:K:gen}
K^\theta((z,y); A \times B)=\iint \1_{A \times B}(z',y') \delta_{\Psi_y^\theta(z)}(dz') G^\theta(\Pi_p(z); dy')\eqsp.
\end{equation}
Note also that, by itself, the process $\{Z_k\,:\, k\in\zsetp\}$ is a Markov chain on $(\Zset,\Zsigma)$ with transition kernel $R^\theta$ satisfying, for all $z\in\Zset$ and $A\in\Zsigma$,

\begin{equation}\label{eq:def:kernel:R:gen}
R^\theta(z;A)=\int \1_A(\Psi^\theta_{y}(z))H^\theta(z;\rmd y)=\int \1_A(\Psi^\theta_{y}(z))G^\theta(\Proj{p}{z};\rmd y)\eqsp.
\end{equation}

Nevertheless, to apply known results to this embedded \acrshort{odm}$(1,1)$, some generalizations are needed. First we assume that $(\Yset,\Ymet)$ is a locally compact, complete and separable space equipped with the metric $\Ymet$. Then, since $(\Xset,\Xmet)$ is a locally compact, complete and separable space, this implies that the space $(\Zset,\Zmet)$ is also locally compact, complete and separable with the metric $\Zmet$ appropriately defined as a function of $\Xmet$ and $\Ymet$ so that $\Zsigma$ is the associated Borel $\sigma$-field. Throughout this {paper}, let us assume the following ergodicity assumption.

\begin{hyp}{AG}
\item\label{assum:gen:identif:unique:pi:gen} For all $\theta\in\Theta$, the
  transition kernel $K^\theta$ of the complete chain admits a unique stationary
  distribution $\pi^\theta$ on $\Zset\times\Yset$.
\end{hyp}
With this assumption, we can now define $\tilde\PP^{\theta}$. The following
notation and
definitions will be used throughout the {paper}.
\begin{definition}\label{def:equi:theta:gen:od}


Under Assumption~\ref{assum:gen:identif:unique:pi:gen}, we denote by $\pi_1^\theta$
and $\pi_2^\theta$
the marginal distributions of $\pi^\theta$ on $\Zset$ and $\Yset$, respectively,
and by $\PP^\theta$ and
$\tilde{\mathbb{P}}^\theta$ the probability distributions defined respectively as follows.
\begin{enumerate}[label=\alph*)]
\item $\PP^\theta$ denotes the extension of $\mathbb{P}_{\pi^\theta}^\theta$ on the whole line
$(\Zset\times\Yset)^{\zset}$.
\item $\tilde{\mathbb{P}}^\theta$ is the corresponding projection on the component $\Yset^{\zset}$.
\end{enumerate}
We also use the symbols
$\mathbb{E}^\theta$ and $\tilde{\mathbb{E}}^\theta$ to denote the expectations
corresponding to $\PP^\theta$ and $\tilde{\mathbb{P}}^\theta$, respectively.
Moreover, for all $\theta, \theta'\in\Theta$, we write $\theta\sim\theta'$ if
and only if $\tilde{\mathbb{P}}^{\theta}=\tilde{\mathbb{P}}^{\theta'}$. This defines an
equivalence relation on the parameter set $\Theta$ and the corresponding
equivalence class of $\theta$ is denoted by $[\theta]\eqdef\{\theta' \in\Theta:\; \theta\sim\theta'\}$.
\end{definition}
The equivalence relationship $\sim$ was introduced by \cite{leroux:1992} as an
alternative to the classical identifiability condition.



For any probability distribution $\Xinit$ on
$\Zset\times\Yset$, we denote by $\PP_\Xinit^\theta$ the
distribution of the Markov chain $\{(Z_k,Y_k)\,:\,k\in\zsetp\}$ with kernel
$K^\theta$ and initial probability mesure $\Xinit$.


\section{Main Results}\label{sec:main-result:gen:od}
\subsection{Preliminaries}

In well-specified setting, a general result on the consistency of the \gls{mle} for a class of first-order \acrshortpl{odm} has been obtained in \cite{dou:kou:mou:2013}. This result has been later extended in \cite{douc2015handy} to a wider class of first-order \acrshortpl{odm}, but the latter has been derived under a handy set of assumptions which appear to be quite direct and easy when checking. The approach used to establish the convergence of the \gls{mle} $\mlY{z,n}$ in these references and in our contribution is briefly described as follows.

First, we establish that, as the number of
observations $n\to\infty$, the
normalized log-likelihood $\mathsf{L}_{z,n}^\theta$ defined in \eqref{eq:defi:lkdM:gen:od}, for some
well-chosen $z\in\Zset$, can be approximated by
$$
n^{-1} \sum_{k=1}^n \ln p^{\theta}(Y_k|Y_{-\infty:k-1}),
$$
where $p^{\theta}(\cdot|\cdot)$ is a $\tilde\PP^\thv\as$ finite real-valued measurable function defined on $(\Yset^\zset,\Ysigma^{\otimes\zset})$.
To define $p^{\theta}(\cdot|\cdot)$, we usually set, for all
$\chunk{y}{-\infty}{1}\in\Yset^{\zsetn}$, whenever the following limit is well
defined,
  \begin{equation}
    \label{eq:def-p-theta-neq-thv:gen:od}
p^{\theta}\left(y_1\,|\,\chunk{y}{-\infty}{0}\right) =
\begin{cases}
\displaystyle\lim_{m\to\infty}  g^\theta\left(\Proj{p}{\F[\theta]{\chunk{y}{-m}{0}}(z)};y_1\right)&
\text{ if the limit exists,}\\
\infty&\text{ otherwise.}
\end{cases}
\end{equation}
By~\ref{assum:gen:identif:unique:pi:gen}, the observed process $\{Y_k:k\in\zset\}$ is ergodic under
$\tilde\PP^{\thv}$ and provided that
$$
\tilde\PE^\thv\left[\lnp p^{\theta}(Y_1|Y_{-\infty:0})\right]<\infty\eqsp,
$$
 it then follows that
$$
\lim_{n\to\infty}\mathsf{L}_{z,n}^\theta=\tilde\PE^\thv\left[\ln p^{\theta}(Y_1|Y_{-\infty:0})\right]\eqsp,\quad\tilde\PP^\thv\as
$$
Finally, we show that with probability tending to one, the MLE $\mlY{z,n}$
eventually lies in a neighborhood of the set
\begin{equation}\label{eq:def-Theta-star-set:gen:od}
\Theta_\star=\argmax_{\theta\in\Theta}\tilde{\PE}^{\thv}\left[\ln
  p^{\theta}(Y_1|\chunk{Y}{-\infty}{0})\right],
\end{equation}
which only depends on $\thv$, establishing that
\begin{equation}\label{eq:strong-consistency-prelim:gen:od}
\lim_{n\to\infty}\met(\mlY{z,n},\Theta_\star)=0,\quad \tilde\PP^{\theta_\star}\as,
\end{equation}
where $\met$ is the metric endowing the parameter space $\Theta$.

In \cite{douc2015handy}, their easy-to-check conditions also induce that for all $\theta,\thv\in\Theta$, there exists a $\tilde\PP^\thv\as$ finite measurable function $\F[\theta]{\cdot}:\Zset^{\zsetn}\to\Zset$ such that for some appropriate value $z\in\Zset$,
\begin{align}\label{eq:def:psi:infty:gen:od}
\lim_{m\to\infty}\F[\theta]{\chunk{Y}{-m}{0}}(z)&=\F[\theta]{\chunk{Y}{-\infty}{0}},\quad\tilde\PP^\thv\as\\ \label{eq:def:Z1:Psi:gen:od}
Z_1&=\F[\thv]{\chunk{Y}{-\infty}{0}},\quad\PP^\thv\as
\end{align}
and that
\begin{align}\label{eq:prob:cond:gen:od}
p^{\theta}(Y_1|\chunk{Y}{-\infty}{0})=g^\theta\left(\Proj{p}{\F[\theta]{\chunk{Y}{-\infty}{0}}};Y_1\right),\quad\tilde\PP^\thv\as
\end{align}
In addition, it is shown that for all $\theta,\thv\in\Theta$,
   \begin{enumerate}[label=(\roman*)]
  \item\label{item:PthetaNEQthetastar:is:density:gen:od} If $\theta\neq\thv$,
    $y\mapsto p^{\theta}(y|\chunk{Y}{-\infty}{0})$ is a density function  $\tilde\PP^{\theta_\star}\as$
  \item\label{item:PthetaISthetastar:is:density:gen:od}  Under
    $\tilde\PP^{\theta_\star}$, the function $y\mapsto
    p^{\thv}(y|\chunk{Y}{-\infty}{0})$ is the conditional density function
    of $Y_1$ given $\chunk{Y}{-\infty}{0}$.
  \end{enumerate}
Under these same conditions, \cite{douc2014maximizing} further showed that the maximizing set $\Theta_\star$ defined in \eqref{eq:def-Theta-star-set:gen:od} indeed reduces to the equivalence class of $\thv$ through the equivalence relation defined in~\autoref{def:equi:theta:gen:od}. Then, the convergence in \eqref{eq:strong-consistency-prelim:gen:od} yields a so-called \emph{equivalence-class consistency}:
\begin{equation}\label{eq:equi:class:consistency:od}
\lim_{n\to\infty}\met(\mlY{z_1,n},[\thv])=0,\quad \tilde\PP^{\theta_\star}\as,
\end{equation}
for some well-chosen $z_1\in\Zset$. From \eqref{eq:equi:class:consistency:od} the (strong) consistency of the \gls{mle} $\mlY{z,n}$ will immediately follow if one can show that $[\thv]$ reduces to the singleton $\{\thv\}$, which is often referred to as solving \emph{identifiability} problem. In this contribution, this issue is also investigated. We will show that if moreover the probability density kernel $G^\theta$ and the measurable function $\F[\theta]{\chunk{Y}{-\infty}{0}}$ defined in \eqref{eq:def:psi:infty:gen:od} and \eqref{eq:def:Z1:Psi:gen:od} satisfy some certain conditions (similar to the conditions in \cite[Proposition~21]{dou:kou:mou:2013}), then the strong consistency holds.

% define
%     $p^{\theta,\thv}(\cdot|\cdot)$ by setting, for
%   $\tilde\PP^{\thv}$-a.e. $\chunk{y}{-\infty}{0}\in\Yset^{\zsetn}$,
%   \begin{equation}
%     \label{eq:def-p1-od}
% p^{\theta,\thv}\left(y_1\,|\,\chunk{y}{-\infty}{0}\right) =
% \begin{cases}
% &\text{ if
%     $\theta\neq\thv$,}\\
% p_1^\theta(y_1\,|\,\chunk{y}{-\infty}{0})\text{ as defined by~(\ref{eq:p-theta-density})}&\text{ otherwise.}
% \end{cases}
%   \end{equation}


\subsection{Convergence of the MLE}\label{sec:convergence:mle}

We always assume in this part that Assumption~\ref{assum:gen:identif:unique:pi:gen} holds throughout. Note that every \acrshort{odm} of any order $p\ge1$ can be embedded in a first-order \acrshort{odm}. Therefore, the approach used to derive the convergence of the \gls{mle} in the class of first-order models can be applied to the class of higher-order ones up to some necessary adaptations. The following is a list of additional assumptions sufficient for obtaining the convergence of the \gls{mle} for a class of higher-order \acrshortpl{odm}.
 \begin{hyp}{AG}
 \item\label{ass:21-lyapunov:gen:od} There exists a
   function $\bar V:\Zset\to\rsetp$ such that, for all $\theta\in\Theta$, $\pi_1^\theta(
   \bar V)<\infty$.
 \end{hyp}
\begin{remark}
  Assumption~\ref{ass:21-lyapunov:gen:od} is usually obtained as a byproduct of the
  proof of Assumption~\ref{assum:gen:identif:unique:pi:gen}; see
  ~\autoref{sec:ergodicity:gen:od}. It is here stated as an assumption for
  convenience.
\end{remark}
The following set of conditions can readily be checked on  $g^\theta$, $\psi^\theta$ and
$\Psi^\theta$.
\begin{hyp}{BG}
\item\label{assum:continuity:Y:g-theta:gen:od}
For all $y \in\Yset$, the function $(\theta,x) \mapsto g^\theta(x;y)$ is
continuous on $\Theta\times\Xset$.
\item\label{assum:continuity:Y:phi-theta:gen:od}
For all $\chunk{y}{1}{p}\in  \Yset$, the function $(\theta,\chunk{x}{1}{p})
\mapsto \psi^\theta_{\chunk{y}{1}{p}}(\chunk{x}{1}{p})$ is continuous  on $\Theta\times\Xset^p$.
\end{hyp}
The function $\bar V$ appearing
in~\ref{assum:practical-cond:O:gen:od}\ref{item:22or23-barphi-V:gen:od} below is the same one
as in Assumption~\ref{ass:21-lyapunov:gen:od}. Moreover, in this condition and
in what follows, we write  $f\lesssim V$ for a real-valued function $f$ and a nonnegative function $V$
defined on the same space $\Zset$, whenever there exists a positive constant
$c$ such that $|f(z)|\leq c V(z)$ for all $z\in\Zset$.
\begin{hyp}{BG}
\item \label{assum:practical-cond:O:gen:od}
There exist $z_1\in\Zset$, a set $\Zset_1\subseteq\Zset$ such that $\Proj{p}{\Zset_1}$ is closed,
  $\varrho\in(0,1)$, $C\geq0$ and
  measurable functions $\bar\Psi:\Zset\to\rsetp$, $H:\rsetp\to\rsetp$ and
  $\bar\phi:\Yset\to\rsetp$ such that the following assertions hold.
\begin{enumerate}[label=(\roman*)]
\item \label{assum:exist:practical-cond:subX:gen:od}
For all
  $\theta\in\Theta$ and $(z, y)\in\Zset \times \Yset$, $\Proj{p}{\Psi^\theta_y(z)}\in\Zset_1$.
\item\label{assum:momentCond:gen:od} $\displaystyle\sup_{(\theta,z,y)\in\Theta\times\Zset_1\times\Yset}g^\theta(\Proj{p}{z};y)<\infty$.
\item  \label{assum:exist:practical-cond:O:gen:od}
For all
  $\theta\in\Theta$, $n\in\zsetp$, $z\in\Zset$, and $\chunk{y}{1}{n}\in\Yset^n$,
  \begin{equation}
    \label{eq:exist:practical-cond:O:gen:od}
\Zmet\left(\F{\chunk{y}{1}{n}}(z_1),\F{\chunk{y}{1}{n}}(z)\right)\leq \varrho^n \; \bar\Psi(z),
  \end{equation}
\item \label{item:psibar:gen:od} $\bar\Psi$ is locally bounded.
\item\label{item:barphi:bar:phi:gen:od} For all $\theta\in\Theta$ and $y\in\Yset$, $\bar\Psi(\Psi_y^\theta(z_1))\leq\bar\phi(y)$.
% \item For all $\theta,\thv\in\Theta$,
% \begin{equation} \label{eq:moment:psi:theta:bar}
% \tilde\PE^{\thv}\left[\lnp\bar\psi\left(\psi^\theta_{Y_0}(x)\right)\right]<\infty\;.
% \end{equation}
\item\label{item:ineq:loggg:gen:od} For all $\theta\in\Theta$ and $(z,z',y)\in\Zset_1\times\Zset_1\times\Yset$,
\begin{equation}\label{eq:ineq:loggg:gen:od}
\left|\ln\frac{g^\theta(\Proj{p}{z};y)}{g^\theta (\Proj{p}{z'};y)}\right|\leq H(\Zmet(z,z')) \,
\rme^{C \, \left(\Zmet(z_1,z)\vee \Zmet(z_1,z')\right)}\;\bar\phi(y),
\end{equation}
\item \label{item:H:gen:od} $H(u)=O(u)$ as $u\to0$.
\item \label{item:22or23-barphi-V:gen:od} If $C=0$, then, for all $\theta\in\Theta$,
\begin{equation}
  \label{eq:barphi-V:gen:od}
 G^\theta\lnp \bar{\phi} \lesssim \bar V;
\end{equation}
otherwise, for all $\theta\in\Theta$,
\begin{equation}
  \label{eq:barphi-strong-V:gen:od}
  G^\theta\bar{\phi}\lesssim \bar V.
\end{equation}
\end{enumerate}
\end{hyp}
Let us now state the equivalence-class consistency of the \gls{mle} for the higher-order \acrshortpl{odm}. The proof can be adapted and easily follows from \cite[Theorem~1]{douc2015handy} and is thus omitted here.

\begin{theorem}\label{thm:convergence-main:gen:od}
  Assume that~\ref{assum:gen:identif:unique:pi:gen}, \ref{ass:21-lyapunov:gen:od},
  \ref{assum:continuity:Y:g-theta:gen:od}, \ref{assum:continuity:Y:phi-theta:gen:od} and \ref{assum:practical-cond:O:gen:od} hold.
  Then, letting $z_1\in\Zset$ as in~\ref{assum:practical-cond:O:gen:od}, the
  convergence~\eqref{eq:equi:class:consistency:od} of the \gls{mle} holds.
\end{theorem}

%\begin{remark} \label{rem:misspecified:gen:od}
%As noticed in \cite{dou:kou:mou:2013}, the techniques used to prove
%\autoref{thm:convergence-main:gen:od} also apply in the misspecified case, where $Y$
%is not distributed according to $\tilde\PP^\thv$. We do not pursue in this
%direction in this contribution.
%\end{remark}

The \emph{strong} consistency of the \gls{mle} follows from the following theorem.
\begin{theorem}\label{thm:strong:consistency:gen:od}
 Assume that~\ref{assum:gen:identif:unique:pi:gen}, \ref{ass:21-lyapunov:gen:od},
  \ref{assum:continuity:Y:g-theta:gen:od}, \ref{assum:continuity:Y:phi-theta:gen:od} and \ref{assum:practical-cond:O:gen:od} hold. Suppose in addition that
\begin{enumerate}[label=(\alph*)]
 \item \label{item:ident:G:gen:od}  for all $\theta=(\vartheta,r), \theta'=(\vartheta',r')\in\Theta$ and $x,x'\in\Xset$,
 \begin{equation*}
 G^\theta(x;\cdot) = G^{\theta'}(x';\cdot)\quad\text{implies}\quad r=r' \quad\text{and}\quad x=x',
 \end{equation*}
 \item \label{item:ident:psi:gen:od}for all $\theta=(\vartheta,r)$, $\thv=(\vartheta_\star, r_\star)\in\Theta$, $\tilde\PP^\thv\as$,
 \begin{equation*}
 \Proj{p}{\F[(\vartheta,r)]{\chunk{Y}{-\infty}{0}}}=\Proj{p}{\F[(\vartheta_\star,r)]{\chunk{Y}{-\infty}{0}}}\quad\text{implies}\quad \vartheta=\vartheta_\star,
 \end{equation*}
 \end{enumerate}
where $\F[\theta]{\chunk{Y}{-\infty}{0}}$ is defined by \eqref{eq:def:psi:infty:gen:od} and \eqref{eq:def:Z1:Psi:gen:od}. Then, letting $z_1\in\Zset$ as in~\ref{assum:practical-cond:O:gen:od},
 \begin{equation}\label{eq:strong:consistency:od}
\lim_{n\to\infty}\met(\mlY{z_1,n},\thv)=0,\quad \tilde\PP^{\theta_\star}\as
\end{equation}
\end{theorem}

 \begin{proof}
 From~\autoref{thm:convergence-main:gen:od}, we have $\Theta_\star=[\thv]$, where $\Theta_\star$ is given in \eqref{eq:def-Theta-star-set:gen:od}. Now let $\theta=(\vartheta,r)\in[\thv]$ and write $\thv=(\vartheta_\star, r_\star)$. The Kulback-Leibler divergence implies, $\tilde\PP^\thv\as$,
 \begin{equation*}
 G^{(\vartheta,r)}\left(\Proj{p}{\F[(\vartheta,r)]{\chunk{Y}{-\infty}{0}}};\cdot\right)=G^{(\vartheta_\star, r_\star)}\left(\Proj{p}{\F[(\vartheta_\star, r_\star)]{\chunk{Y}{-\infty}{0}}};\cdot\right).
 \end{equation*}
 Then from \ref{item:ident:G:gen:od}, we obtain
 $$r=r_\star$$
 and
 \begin{equation*}
 \Proj{p}{\F[(\vartheta,r)]{\chunk{Y}{-\infty}{0}}}=\Proj{p}{\F[(\vartheta_\star, r_\star)]{\chunk{Y}{-\infty}{0}}},\quad\tilde\PP^\thv\as
 \end{equation*}
 From \ref{item:ident:psi:gen:od}, we also have $\vartheta=\vartheta_\star$. Thus the proof follows.
 \end{proof}


\subsection{Asymptotic Normality of the MLE}\label{sec:clt:gen:od}
In this section, we treat a special but an important case of \acrshortpl{odm} where the space $\Zset$ is a subset of a finite-dimensional vector space; $\Theta$ is a compact subset of $\rset^d$, for some positive integer $d$; and the function $\Psi_{y}^\theta(z)$ defined in \eqref{eq:def:psi:gen:od} admits the following form: for all $(z,y,\theta)\in\Zset\times\Yset\times\Theta$,
\begin{equation}\label{eq:def:psi:linear:od}
\Psi_{y}^\theta(z)=a(\theta,y)z+b(\theta,y),
\end{equation}
where $a(\theta,y)$ and $b(\theta,y)$ are two matrices with appropriate dimensions, and for all $\theta\in\Theta$, $y\mapsto a(\theta,y)$ and $y\mapsto b(\theta,y)$ are measurable maps on $(\Yset,\Ysigma)$. The underlying model is assumed to be well-specified and  $\thv\in\Theta$ denotes the true parameter. As in~\autoref{sec:convergence:mle}, Assumption~\ref{assum:gen:identif:unique:pi:gen} is supposed to hold throughout this section. Assumption~\ref{assum:phi:linear:od} below guarantees the existence of $\F[\theta]{\cdot}$ satisfying~\eqref{eq:def:psi:infty:gen:od} and \eqref{eq:def:Z1:Psi:gen:od}, and its differentiability on $\Theta$. In what follows, we  denote by $|\cdot|$ a vector norm on $\rset^d$ and by $\|\cdot\|$ a matrix norm.
\begin{hyp}{CG}
\item\label{assum:phi:linear:od}
We have the following.
\begin{enumerate}[label=(\roman*)]
\item\label{item:differentiability:a:b:od}
For all $y\in\Yset$, $\theta\mapsto a(\theta,y)$ and $\theta\mapsto b(\theta,y)$ are twice continuously differentiable on $\Theta$.
\item\label{item:ineq:prod:rho:n:od}
There exist constants $C>0$ and $\rho\in(0,1)$ such that for all $\theta\in\Theta$ and $\chunk{y}{1}{n}\in\Yset$, $$\left\|\prod_{\ell=1}^{n}a(\theta,y_{\ell})\right\|\le C\rho^n.$$
\item\label{item:finite:sup:b:od}
$\tilde\PE^\thv[\sup_{\theta\in\Theta}\|b(\theta,Y_{1})\|]<\infty$.
\item\label{item:ineq:first:diff-a-b:finite:od}
For all $i\in\{1,\ldots,d\}$,
$$\tilde\PE^\thv\left[\sup_{\theta\in\Theta}\left\|\frac{\partial a(\theta,Y_{1})}{\partial\theta_i}\right\|\right]+\tilde\PE^\thv\left[\sup_{\theta\in\Theta}\left\|\frac{\partial b(\theta,Y_{1})}{\partial\theta_i}\right\|\right]<\infty.$$
\item\label{item:ineq:second:diff-a-b:finite:od}
For all $i, j\in\{1,\ldots,d\}$,
$$\tilde\PE^\thv\left[\sup_{\theta\in\Theta}\left\|\frac{\partial^2a(\theta,Y_{1})}{\partial\theta_i\partial\theta_j}\right\|\right]+\tilde\PE^\thv\left[\sup_{\theta\in\Theta}\left\|\frac{\partial^2b(\theta,Y_{1})}{\partial\theta_i\partial\theta_j}\right\|\right]<\infty.$$
\end{enumerate}
\end{hyp}
%\item\label{item:finite:sup:b:od}
%$\tilde\PE^\thv[\lnp\sup_{\theta\in\Theta}\|b(\theta,Y_{1})\|]<\infty$.
%\item\label{item:ineq:first:diff-a-b:finite:od}
%For all $i\in\{1,\ldots,d\}$,
%$$\tilde\PE^\thv\left[\lnp\sup_{\theta\in\Theta}\left\|\frac{\partial a(\theta,Y_{1})}{\partial\theta_i}\right\|\right]+\tilde\PE^\thv\left[\lnp\sup_{\theta\in\Theta}\left\|\frac{\partial b(\theta,Y_{1})}{\partial\theta_i}\right\|\right]<\infty.$$
%\item\label{item:ineq:second:diff-a-b:finite:od}
%For all $i, j\in\{1,\ldots,d\}$,
%$$\tilde\PE^\thv\left[\lnp\sup_{\theta\in\Theta}\left\|\frac{\partial^2a(\theta,Y_{1})}{\partial\theta_i\partial\theta_j}\right\|\right]+\tilde\PE^\thv\left[\lnp\sup_{\theta\in\Theta}\left\|\frac{\partial^2b(\theta,Y_{1})}{\partial\theta_i\partial\theta_j}\right\|\right]<\infty.$$
%\end{enumerate}
%\end{hyp}

\begin{lemma}\label{lem:exist:deriv:psi:od}
Assume \ref{assum:gen:identif:unique:pi:gen} and \ref{assum:phi:linear:od}. Then for all $\theta\in\Theta$, there exists a $\tilde\PP^\thv\as$ finite measurable function $\F[\theta]{\cdot}:\Zset^{\zsetn}\to\Zset$ satisfying \eqref{eq:def:psi:infty:gen:od} and \eqref{eq:def:Z1:Psi:gen:od} for all $z\in\Zset$. The function $\theta\mapsto\F{\chunk{Y}{-\infty}{0}}$ is $\tilde\PP^\thv\as$ twice continuously differentiable on $\Theta$ and uniformly on $\Theta$, we have $\tilde\PP^\thv\as$,
\begin{align}\label{eq:def:first:diif:psi:infty:gen:od}
\frac{\partial\F[\theta]{\chunk{Y}{-\infty}{0}}}{\partial\theta}&=\lim_{m\to\infty}\frac{\partial\F[\theta]{\chunk{Y}{-m}{0}}(z)}{\partial\theta} \\ \label{eq:def:second:diif:psi:infty:gen:od}
\frac{\partial^2\F[\theta]{\chunk{Y}{-\infty}{0}}}{\partial\theta^T\partial\theta}&=\lim_{m\to\infty}\frac{\partial^2\F[\theta]{\chunk{Y}{-m}{0}}(z)}{\partial\theta^T\partial\theta}.
\end{align}
Moreover,
\begin{align}
\label{eq:finite:F1:gen:od}
&\PEY\left[\sup_{\theta\in\Theta}\left\|\F{\chunk{Y}{-\infty}{0}}\right\|\right]<\infty, \\ \label{eq:finite:F2:gen:od}
&\PEY\left[\sup_{\theta\in\Theta}\left\|\frac{\partial\F[\theta]{\chunk{Y}{-\infty}{0}}}{\partial\theta}\right\|\right]<\infty, \\ \label{eq:finite:F3:gen:od}
&\PEY\left[\sup_{\theta\in\Theta}\left\|\frac{\partial^2\F[\theta]{\chunk{Y}{-\infty}{0}}}{\partial\theta^T\partial\theta}\right\|\right]<\infty.
\end{align}
\end{lemma}
\begin{proof}
The proof is postponed to \autoref{sec:proof:lem:deriv:psi:od:gen} for convenience.
\end{proof}

\begin{hyp}{CG}
\item\label{assum:differentiability:g:od} For all $y\in\Yset$, the function $(\theta,x)\mapsto g^\theta(x;y)$ twice continuously differentiable on $\Theta\times\Xset$.
\end{hyp}

For stating further assumptions, the following notation may be needed. For all $k\in\zsetpnz$, $\theta\in\theta$ and $z\in\Zset$, let
$$\ell_{z,k}^\theta\eqdef\ln g^\theta\left(\Proj{p}{\F{\chunk{Y}{1}{k-1}}(z)} ;Y_k\right).$$
Then for all $n\in\zsetpnz$, it follows that $\lkdM[z,n]{}[\theta]$ defined by \eqref{eq:defi:lkdM:gen:od} can be rewritten as
\begin{equation}\label{eq:Lz:ellz:gen:od}
\lkdM[z,n]{}[\theta]=n^{-1}\sum_{k=1}^n\ell_{z,k}^\theta
\end{equation}
For all $k\in\zsetpnz$, $\theta\in\theta$ and $z\in\Zset$, let
$$\ell_{k}^\theta\eqdef\ln g^\theta\left(\Proj{p}{\F{\chunk{Y}{-\infty}{k-1}}} ;Y_k\right),$$
which is $\tilde\PP^\thv\as$ well defined by \autoref{lem:exist:deriv:psi:od}. Note that from \eqref{eq:prob:cond:gen:od}, we have for all $k\in\zsetpnz$, $\theta\in\theta$, $\ell_{k}^\theta=\ln p^{\theta}(Y_k|\chunk{Y}{-\infty}{k-1})$. Now for all $n\in\zsetpnz$ and $\theta\in\Theta$, define
\begin{equation}\label{eq:L:ell:gen:od}
\lkdM[n]{}[\theta]\eqdef n^{-1}\sum_{k=1}^n\ell_{k}^\theta.
\end{equation}

\begin{remark}\label{rem:regularity:ell:gen:od}
Note that by \autoref{lem:exist:deriv:psi:od},~\ref{assum:gen:identif:unique:pi:gen},
~\ref{assum:phi:linear:od}  and
\ref{assum:differentiability:g:od} imply that the functions
$\theta\mapsto\ell_{z,k}^\theta$ and $\theta\mapsto\ell_{k}^\theta$ are twice
continuously differentiable on some neighborhood $\mathcal{V}(\thv)$ of $\thv$,
$\tilde\PP^\thv\as$
\end{remark}
Assumptions~\ref{assum:gen:identif:unique:pi:gen}, ~\ref{assum:phi:linear:od} and \ref{assum:differentiability:g:od} together with the following set of assumptions, \ref{assum:interior:theta:star}--\ref{item:inverse:fisher:gen:od}, are sufficient to yield the asymptotic normality of $\mlY{z,n}$ for arbitrary $z\in\Zset$.

\begin{hyp}{CG}
\item\label{assum:interior:theta:star} The true parameter $\thv$ lies within the interior of $\Theta$.
\item\label{assum:strong:consistent}
The strong consistency holds, that is, $\lim_{n\to\infty}\mlY{z,n}=\thv,\quad \tilde\PP^\thv\as$
\item\label{item:ctl:limit:od:gen}
There exist a constant $\alpha>1/2$ and a $\tilde\PP^\thv\as$ finite random variable $C>0$ such that for all $k\ge 1$,
$$
\sup_{\theta\in\mathcal{V}(\thv)}\left|\frac{\partial\ell_{z,k}^\theta}{\partial\theta}-\frac{\partial\ell_{k}^\theta}{\partial\theta}\right|\le Ck^{-\alpha},\quad\tilde\PP^\thv\as,
$$
where $\mathcal{V}(\thv)$ is a neighborhood of $\thv$.
%There exists a constant $\varrho\in[0,1)$ such that \mo{can be $\le Ck^{-\alpha}$, $\alpha>1/2$, leaving $\varrho$}
%$$
%\lim_{k\to\infty}\sup_{\theta\in\mathcal{V}(\thv)}\varrho^{-k}\left|\frac{\partial\ell_{z,k}^\theta}{\partial\theta}-\frac{\partial\ell_{k}^\theta}{\partial\theta}\right|=0,\quad\tilde\PP^\thv\as
%$$
\item\label{item:finte:mement:gen:od} The exists a neighborhood $\mathcal{V}(\thv)$ of $\thv$ such that
\begin{enumerate}[label=(\roman*)]
\item\label{item:1:finte:moment:od:gen}
$\tilde\PE^\thv\left[\sup_{\theta\in\mathcal{V}(\thv)}\left|\frac{\partial\ell_{1}^\theta}{\partial\theta}\right|^2\right]<\infty$.
\item\label{item:2:finte:moment:od:gen}
$\tilde\PE^\thv\left[\sup_{\theta\in\mathcal{V}(\thv)}\left\|\frac{\partial^2\ell_{1}^\theta}{\partial\theta^T\partial\theta}\right\|\right]<\infty$.
\item\label{item:3:finte:moment:od:gen}
$\tilde\PE^\thv\left[\sup_{\theta\in\mathcal{V}(\thv)}\left|\frac{\partial}{\partial\theta}g^\theta\left(\Proj{p}{\F{\chunk{Y}{-\infty}{0}}} ;Y_1\right)\right|\right]<\infty$.
\item\label{item:4:finte:moment:od:gen}
$\tilde\PE^\thv\left[\sup_{\theta\in\mathcal{V}(\thv)}\left\|\frac{\partial^2}{\partial\theta^T\partial\theta}g^\theta\left(\Proj{p}{\F{\chunk{Y}{-\infty}{0}}} ;Y_1\right)\right\|\right]<\infty$.
\end{enumerate}
\item\label{item:inverse:fisher:gen:od}
The matrix $\Sigma_\star=-\tilde\PE^\thv\left[\frac{\partial^2\ell_{1}^\thv}{\partial\theta^T\partial\theta}\right]$ is invertible.
\end{hyp}
%\begin{remark}
%The neighborhoods of $\thv$ in the above assumptions can be all different. Nevertheless, these assumptions still hold with $\mathcal{V}(\thv)$ taken to be the intersection of these neighborhoods, which is again a neighborhood of $\thv$. Thus without loss of generality, we can assume that all the $\mathcal{V}(\thv)$ above agree.
%\end{remark}


Note that if Assumptions ~\ref{assum:gen:identif:unique:pi:gen},
~\ref{assum:phi:linear:od} and \ref{assum:differentiability:g:od} hold, then by \autoref{rem:regularity:ell:gen:od}, $\theta\mapsto\lkdM[z,n]{}[\theta]$ and $\theta\mapsto\lkdM[n]{}[\theta]$ respectively given by \eqref{eq:Lz:ellz:gen:od} and \eqref{eq:L:ell:gen:od} are twice
continuously differentiable on some neighborhood $\mathcal{V}(\thv)$ of $\thv$,
$\tilde\PP^\thv\as$ If in addition Assumption~\ref{item:ctl:limit:od:gen} holds, the following fact is obtained.

\begin{lemma}\label{lem:help:close:like:od:gen}
Assume ~\ref{assum:gen:identif:unique:pi:gen},
~\ref{assum:phi:linear:od}, ~\ref{assum:differentiability:g:od} and \ref{item:ctl:limit:od:gen}, then
$$
\lim_{n\to\infty}\sup_{\theta\in\mathcal{V}(\thv)}n^{1/2}\left|\frac{\partial\lkdM[z,n]{}[\theta]}{\partial\theta}- \frac{\partial\lkdM[n]{}[\theta]}{\partial\theta}\right|=0,\quad\tilde\PP^\thv\as,
$$
\end{lemma}
\begin{proof}
We have from \ref{item:ctl:limit:od:gen}, $\tilde\PP^\thv\as$,
\begin{align*}
\begin{split}
\sup_{\theta\in\mathcal{V}(\thv)}n^{1/2}\left|\frac{\partial\lkdM[z,n]{}[\theta]}{\partial\theta}- \frac{\partial\lkdM[n]{}[\theta]}{\partial\theta}\right|&\le\frac{1}{n^{1/2}}\sum_{k=1}^n\sup_{\theta\in\mathcal{V}(\thv)}\left|\frac{\partial\ell_{z,k}^\theta}{\partial\theta}-\frac{\partial\ell_{k}^\theta}{\partial\theta}\right|\le\frac{C}{n^{1/2}}\sum_{k=1}^n\frac{1}{k^\alpha}.
\end{split}
\end{align*}
To complete, observe that the rightmost term of the above inequalities converges to $0$ as $n\to\infty$ whenever $\alpha>1/2$.
\end{proof}


\begin{theorem}\label{thm:asym:normal:od:gen}
Assume that~\ref{assum:gen:identif:unique:pi:gen} and ~\ref{assum:phi:linear:od}--\ref{item:inverse:fisher:gen:od} hold. Then, for any $z\in\Zset$,
\begin{equation}
\sqrt{n}(\mlY{z,n}-\thv)\rightsquigarrow_{\thv}\mathcal{N}(0,\Sigma_\star^{-1})\eqsp,
\end{equation}
where the symbol $\rightsquigarrow_{\thv}$ means weak convergence under $\tilde\PP^\thv$, and $\mathcal{N}(0,\Sigma_\star^{-1})$ stands for the centered Gaussian distribution with covariance matrix $\Sigma_\star^{-1}$.
\end{theorem}


\begin{proof}
From~\autoref{lem:help:close:like:od:gen}, we have
\begin{align*}
\lim_{n\to\infty}n^{1/2}\sup_{\theta\in\mathcal{V}(\thv)}\left|\frac{\partial\lkdM[z,n]{}[\theta]}{\partial\theta}-\frac{\partial\lkdM[n]{}[\theta]}{\partial\theta}\right|=0,\quad\tilde\PP^\thv\as
\end{align*}
Since for sufficiently large $n$, $\mlY{z,n}\in\mathcal{V}(\thv)$ and $\frac{\partial\lkdM[z,n]{}[\mlY{z,n}]}{\partial\theta}=0$, then
\begin{align*}
n^{1/2}\frac{\partial\lkdM[n]{}[\mlY{z,n}]}{\partial\theta}=n^{1/2}\frac{\partial\lkdM[z,n]{}[\mlY{z,n}]}{\partial\theta}+o_P(1)=o_P(1).
\end{align*}
Note that we also have,
\begin{align*}
n^{1/2}\frac{\partial \lkdM[n]{}[\mlY{z,n}]}{\partial\theta}=n^{1/2}\frac{\partial\lkdM[n]{}[\thv]}{\partial\theta}+\left(\frac{\partial^2\lkdM[n]{}[\thv]}{\partial\theta^T\partial\theta}+\varepsilon_n\right)n^{1/2}(\mlY{z,n}-\thv),
\end{align*}
where
\begin{equation}\label{eq:epsilon:n:gen:od}
\varepsilon_n=\int_{0}^{1}\left(\frac{\partial^2 \lkdM[n]{}[\thv+t(\mlY{z,n}-\thv)]}{\partial\theta^T\partial\theta}-\frac{\partial^2\lkdM[n]{}[\thv]}{\partial\theta^T\partial\theta}\right)\rmd t.
\end{equation}
Thus,
\begin{align*}
n^{1/2}\frac{\partial \lkdM[n]{}[\thv]}{\partial\theta}=-\left(\frac{\partial^2\lkdM[n]{}[\thv]}{\partial\theta^T\partial\theta}+\varepsilon_n\right)n^{1/2}(\mlY{z,n}-\thv)+o_P(1).
\end{align*}
%Note that we also have,
%\begin{align*}
%n^{1/2}\frac{\partial \lkdM[n]{}[\mlY{z,n}]}{\partial\theta}=n^{1/2}\frac{\partial\lkdM[n]{}[\thv]}{\partial\theta}+(-\Sigma_\star+\varepsilon_n)n^{1/2}(\mlY{z,n}-\thv),
%\end{align*}
%where
%\begin{equation*}
%\varepsilon_n=\int_{0}^{1}\left(\frac{\partial^2 \lkdM[n]{}[\thv+t(\mlY{z,n}-\thv)]}{\partial\theta^T\partial\theta}+\Sigma_\star\right)\rmd t.
%\end{equation*}
%Thus,
%\begin{align*}
%n^{1/2}\frac{\partial \lkdM[n]{}[\thv]}{\partial\theta}=(\Sigma_\star-\varepsilon_n)n^{1/2}(\mlY{z,n}-\thv)+o_P(1).
%\end{align*}
Note that from \ref{item:finte:mement:gen:od}\ref{item:2:finte:moment:od:gen} and by ergodicity, Birkhoff ergodic theorem applies and we have
$$
\lim_{n\to\infty}\frac{\partial^2\lkdM[n]{}[\thv]}{\partial\theta^T\partial\theta}=\lim_{n\to\infty}n^{-1}\sum_{k=1}^n\frac{\partial^2\ell_k^\thv}{\partial\theta^T\partial\theta}=-\Sigma_\star,\quad\tilde\PP^\thv\as
$$
Thus to complete the proof, we only need to show that
\begin{enumerate}[label=(\alph*)]
\item\label{item:martingale:diff:L}
$n^{1/2}\frac{\partial \lkdM[n]{}[\thv]}{\partial\theta}\rightsquigarrow_{\thv}\mathcal{N}(0,\Sigma_\star)$ and
\item\label{item:small:O:epsilon}
$\varepsilon_n=o_P(1)$.
\end{enumerate}
Let us now we show \ref{item:martingale:diff:L}. Note that we can write
\begin{align*}
n^{1/2}\frac{\partial \lkdM[n]{}[\thv]}{\partial\theta}=n^{-1/2}\sum_{k=1}^n\frac{\partial\ell_{k}^\thv}{\partial\theta}.
\end{align*}
Let $\xi_{k}=\frac{\partial\ell_{k}^\thv}{\partial\theta}$, $k\in\zsetp$. From~\ref{item:finte:mement:gen:od}\ref{item:3:finte:moment:od:gen}, it readily follows that
\begin{align*}
\tilde\PE^\thv\left[\left.\xi_{k}\right|\chunk{Y}{-\infty}{k-1}\right]&=\int_{\Yset}\frac{\partial g^\thv\left(\Proj{p}{\F[\thv]{\chunk{Y}{-\infty}{k-1}}} ;y\right)}{\partial\theta}\nu(\rmd y) \\
&=\left.\frac{\partial}{\partial\theta}\int_{\Yset} g^\theta\left(\Proj{p}{\F[\theta]{\chunk{Y}{-\infty}{k-1}}} ;y\right)\nu(\rmd y)\right|_{\theta=\thv}\\
&=\left.\frac{\partial}{\partial\theta}1\,\right|_{\theta=\thv}=0,
\end{align*}
so that the sequence $\sequence{\xi}[k][\zsetp]$ is a stationary and ergodic Martingale difference with respect to $\chunk{\mathcal{F}}{-\infty}{k}=\sigma(\chunk{Y}{-\infty}{k})$ in $L_2(\tilde\PP^\thv)$. By \ref{item:finte:mement:gen:od} and \ref{item:inverse:fisher:gen:od}, its covariance matrix is equal to $\Sigma_\star$. To see this, observe that $\tilde\PP^\thv\as$,
\begin{align*}
&\tilde\PE^\thv\left[\left.\xi_{k}\xi_k^T\right|\chunk{\mathcal{F}}{-\infty}{k-1}\right] \\
&=\int_{\Yset}\frac{1}{g^\thv\left(\psi_k^\thv ;y\right)}\left(\frac{\partial g^\thv\left( \psi_k^\thv;y\right)}{\partial\theta}\right)\left(\frac{\partial g^\thv\left(\psi_k^\thv ;y\right)}{\partial\theta}\right)^T\nu(\rmd y) \\
&=\int_{\Yset} \left(\frac{\partial^2  g^\thv\left(\psi_k^\thv ;y\right)}{\partial\theta^T\partial\theta}-\frac{\partial^2 \ln g^\thv\left(\psi_k^\thv ;y\right)}{\partial\theta^T\partial\theta}g^\thv\left(\psi_k^\thv ;y\right)\right)\nu(\rmd y),
\end{align*}
with $\psi_k^\thv \eqdef \Proj{p}{\F[\thv]{\chunk{Y}{-\infty}{k-1}}}$. By  \ref{item:finte:mement:gen:od}\ref{item:4:finte:moment:od:gen}, we have
\begin{align*}
\int_{\Yset} \frac{\partial^2  g^\thv\left(\psi_k^\thv ;y\right)}{\partial\theta^T\partial\theta}=0,\quad\tilde\PP^\thv\as
\end{align*}
Since $\tilde\PP^\thv\as$,
\begin{align*}
\tilde\PE^\thv\left[\left.\frac{\partial^2\ell_k^\thv}{\partial\theta^T\partial\theta}\right|\chunk{\mathcal{F}}{-\infty}{k-1}\right]=\int_{\Yset} \frac{\partial^2 \ln g^\thv\left(\psi_k^\thv ;y\right)}{\partial\theta^T\partial\theta}g^\thv\left(\psi_k^\thv ;y\right)\nu(\rmd y)\eqsp,
\end{align*}
it then follows that
\begin{align*}
\tilde\PE^\thv\left[\left.\xi_{k}\xi_k^T\right|\chunk{\mathcal{F}}{-\infty}{k-1}\right]=-\tilde\PE^\thv\left[\left.\frac{\partial^2\ell_k^\thv}{\partial\theta^T\partial\theta}\right|\chunk{\mathcal{F}}{-\infty}{k-1}\right],\quad\tilde\PP^\thv\as
\end{align*}
Taking the expectation on both sides, then using tower property and using ~\ref{item:inverse:fisher:gen:od}, the claim thus follows.
By ergodicity and from \ref{item:finte:mement:gen:od}\ref{item:1:finte:moment:od:gen}, we have
\begin{align*}
\lim_{n\to\infty}n^{-1}\sum_{k=1}^n\tilde\PE^\thv\left[\left.\xi_{k}\xi_{k}^T\right|\chunk{\mathcal{F}}{-\infty}{k-1}\right]=\Sigma_\star,\quad\tilde\PP^\thv\as
\end{align*}
Moreover, for $\epsilon>0$,
\begin{align*}
\lim_{n\to\infty}n^{-1}&\left\|\sum_{k=1}^n\tilde\PE^\thv\left[\left.\xi_{k}\xi_{k}^T\1(|\xi_{k}|>\epsilon\sqrt{n})\right|\chunk{\mathcal{F}}{-\infty}{k-1}\right]\right\| \\
&\le \lim_{n\to\infty}n^{-1}\sum_{k=1}^n\tilde\PE^\thv\left[\left.\left|\xi_{k}\right|^2\1(|\xi_{k}|>\epsilon\sqrt{n})\right|\chunk{\mathcal{F}}{-\infty}{k-1}\right] \\
&\le \lim_{n\to\infty}n^{-1}\sum_{k=1}^n\tilde\PE^\thv\left[\left.\left|\xi_{k}\right|^2\1(|\xi_{k}|>M)\right|\chunk{\mathcal{F}}{-\infty}{k-1}\right] \\
&=\tilde\PE^\thv\left[\left|\xi_{1}\right|^2\1(|\xi_{1}|>M)\right], \quad\tilde\PP^\thv\as,
\end{align*}
where $M>0$. Since $M$ is arbitrary, letting $M\to\infty$, we have
\begin{align*}
\lim_{n\to\infty}n^{-1}\sum_{k=1}^n\tilde\PE^\thv\left[\left.\xi_{k}\xi_{k}^T\1(|\xi_{k}|>\epsilon\sqrt{n})\right|\chunk{\mathcal{F}}{-\infty}{k-1}\right]=0, \quad\tilde\PP^\thv\as
\end{align*}
Thus by martingale central limit theorem, see, for instance, \cite{hall:heyde:1980},
$$n^{1/2}\frac{\partial\lkdM[n]{}[\thv]}{\partial\theta}=n^{-1/2}\sum_{k=1}^n\xi_{k}\rightsquigarrow_{\thv}\mathcal{N}(0,\Sigma_\star).$$
Now it remains to show \ref{item:small:O:epsilon}.  Let $\rho>0$ such that
$\ball{\thv}{\rho}\subset\mathcal{V}(\thv)$. Since $\tilde\PP^\thv\as$,
$\mlY{z,n}\to\thv$ as $n\to\infty$, then there exists a $\tilde\PP^\thv\as$
finite integer $N_0$ such that for all $n\ge N_0$, $\tilde\PP^\thv\as$,
$\thv+t(\mlY{z,n}-\thv)\in\ball{\thv}{\rho}$. Then from
\eqref{eq:epsilon:n:gen:od}, $\tilde\PP^\thv\as$, for all  $n\geq N_0$,
\begin{align*}
\|\varepsilon_n\| &\le\int_{0}^{1}\left\|\frac{\partial^2 \lkdM[n]{}[\thv+t(\mlY{z,n}-\thv)]}{\partial\theta^T\partial\theta}-\frac{\partial^2\lkdM[n]{}[\thv]}{\partial\theta^T\partial\theta}\right\|\rmd t \\
&\le \sup_{\theta\in\ball{\thv}{\rho}}\left\|\frac{\partial^2 \lkdM[n]{}[\theta]}{\partial\theta^T\partial\theta}-\frac{\partial^2\lkdM[n]{}[\thv]}{\partial\theta^T\partial\theta}\right\| \\
&\le n^{-1}\sum_{k=1}^n\sup_{\theta\in\ball{\thv}{\rho}}\left\|\frac{\partial^2 \ell_k^\theta}{\partial\theta^T\partial\theta}-\frac{\partial^2\ell_k^\thv}{\partial\theta^T\partial\theta}\right\|\eqsp.
\end{align*}
By \ref{item:finte:mement:gen:od}\ref{item:2:finte:moment:od:gen},
$\ball{\thv}{\rho}\subset\mathcal{V}(\thv)$ and Birkhoff ergodic theorem, we get that
\begin{align*}
\limsup_{n\to\infty}\|\varepsilon_n\|&\le\tilde\PE^\thv\left[\sup_{\theta\in\ball{\thv}{\rho}}\left\|\frac{\partial^2
      \ell_1^\theta}{\partial\theta^T\partial\theta}-\frac{\partial^2\ell_1^\thv}{\partial\theta^T\partial\theta}\right\|\right],\quad
\tilde\PP^\thv\as
\end{align*}
Here $\rho$ is an arbitrary positive number, provided that
$\ball{\thv}{\rho}\subset\mathcal{V}(\thv)$.
By dominated convergence and \autoref{rem:regularity:ell:gen:od}, the right-hand side of the previous display can be made
arbitrarily close to zero by letting $\rho$ tend to zero. Hence we get
\ref{item:small:O:epsilon}, and the proof is complete.
\end{proof}

\begin{remark}\label{rem:hessian:invet:gen:od}
From the proof of \autoref{thm:asym:normal:od:gen}, it is also shown that
\begin{align*}
\Sigma_\star=\tilde\PE^\thv\left[\left(\frac{\partial\ell_{1}^\thv}{\partial\theta}\right)\left(\frac{\partial\ell_{1}^\thv}{\partial\theta}\right)^T\right].
\end{align*}
Thus, to obtain \ref{item:inverse:fisher:gen:od}, that is showing that $\Sigma_\star$ is invertible, it is sufficient to show that for all $\mathbf{a}\in\rset^d$,
\begin{align}\label{eq:rem:linear:independent:d-logpois:gen:od}
\mathbf{a}^T\frac{\partial\ell_{1}^\thv}{\partial\theta}=0,\quad\tilde\PP^\thv\as, \implies \mathbf{a}=0\eqsp.
\end{align}
\end{remark}



%$$
%\f[\theta]{\chunk{y}{-m}{0}}(x)-\f[\theta]{\chunk{y}{-m}{0}}(x')=\left(\prod_{\ell=0}^{m}a(\theta,y_{-\ell})\right)(x-x').
%$$
%This implies,
%\begin{align*}
%&\frac{\partial\f[\theta]{\chunk{y}{-m}{0}}(x)}{\partial\theta}-\frac{\partial\f[\theta]{\chunk{y}{-m}{0}}(x')}{\partial\theta} \\
%&=\sum_{j=0}^m\left[(x-x')^T\left(\prod_{\ell=j+1}^{m}a(\theta,y_{-\ell})\right)^T\otimes\left(\prod_{\ell=0}^{j-1}a(\theta,y_{\ell-j+1})\right)\right]\frac{\partial\,\text{vec}\, a(\theta,y_{-j})}{\partial\theta}.
%\end{align*}
%Then,
%\begin{align*}
%\left\|\frac{\partial\f[\theta]{\chunk{y}{-m}{0}}(x)}{\partial\theta}-\frac{\partial\f[\theta]{\chunk{y}{-m}{0}}(x')}{\partial\theta}\right\|&\le M\rho^m\sum_{j=0}^m\left\|\frac{\partial\,\text{vec}\, a(\theta,y_{-j})}{\partial\theta}\right\||x-x'|
%\end{align*}
%If we assume that
%$$
%\tilde\PE^\thv\left[\lnp\sup_{\theta\in\Theta}\left\|\frac{\partial\,\text{vec}\, a(\theta,Y_{1})}{\partial\theta}\right\|\right]<\infty,
%$$
%and
%$$
%\tilde\PE^\thv\left[\lnp\sup_{\theta\in\Theta}\left\|\frac{\partial\,\text{vec}\, b(\theta,Y_{1})}{\partial\theta}\right\|\right]<\infty,
%$$
%then there exist a $\tilde\PP^\thv$-almost surely finite random variable $\tilde M$ and a constant $\bar \rho\in(0,1)$ such that for all $x,x'\in\Xset$,
%\begin{align*}
%\left\|\frac{\partial\f[\theta]{\chunk{y}{-m}{0}}(x)}{\partial\theta}-\frac{\partial\f[\theta]{\chunk{y}{-m}{0}}(x')}{\partial\theta}\right\|&\le \tilde M\bar\rho^m|x-x'|
%\end{align*}
%For any $x\in\Xset$, we have
%\begin{align*}
%&\frac{\partial\f[\theta]{\chunk{Y}{-m-1}{0}}(x)}{\partial\theta}-\frac{\partial\f[\theta]{\chunk{Y}{-m}{0}}(x)}{\partial\theta} \\
%&=\frac{\partial}{\partial\theta}\left[\f[\theta]{\chunk{Y}{-m}{0}}(\psi_{Y_{-m-1}}^\theta(x))\right]-\frac{\partial\f[\theta]{\chunk{Y}{-m}{0}}(x)}{\partial\theta}\\
%&=\frac{\partial\f[\theta]{\chunk{Y}{-m}{0}}}{\partial\theta}(\psi_{Y_{-m-1}}^\theta(x))+\frac{\partial\f[\theta]{\chunk{Y}{-m}{0}}}{\partial x}(\psi_{Y_{-m-1}}^\theta(x))\frac{\partial \psi_{Y_{-m-1}}^\theta(x)}{\partial\theta}\\
%&-\frac{\partial\f[\theta]{\chunk{Y}{-m}{0}}(x)}{\partial\theta},
%\end{align*}
%which by.... implies
%\begin{align*}
%&\left\|\frac{\partial\f[\theta]{\chunk{Y}{-m-1}{0}}(x)}{\partial\theta}-\frac{\partial\f[\theta]{\chunk{Y}{-m}{0}}(x)}{\partial\theta}\right\| \\
%&\le\left\|\frac{\partial\f[\theta]{\chunk{Y}{-m}{0}}}{\partial\theta}(\psi_{Y_{-m-1}}^\theta(x))-\frac{\partial\f[\theta]{\chunk{Y}{-m}{0}}(x)}{\partial\theta}\right\|\\
%&+\left\|\prod_{\ell=0}^{m}a(\theta,y_{-\ell})\right\|\,\left\|\frac{\partial \psi_{Y_{-m-1}}^\theta(x)}{\partial\theta}\right\| \\
%&\le \tilde M\bar\rho^m|\psi_{Y_{-m-1}}^\theta(x)-x| \\
%&+C\rho^m\left(\|x^T\otimes I\| \sup_{\theta\in\Theta}\left\|\frac{\partial\,\text{vec}\, a(\theta,Y_{-m-1})}{\partial\theta}\right\|+\sup_{\theta\in\Theta}\left\|\frac{\partial\,\text{vec}\, b(\theta,Y_{-m-1})}{\partial\theta}\right\|\right)\\
%&\le \tilde M\bar\rho^m\left(\beta |x|+\sup_{\theta\in\Theta}|b(\theta,Y_{-m-1})|\right) \\
%&+C\rho^m\left(\|x^T\otimes I\| \sup_{\theta\in\Theta}\left\|\frac{\partial\,\text{vec}\, a(\theta,Y_{-m-1})}{\partial\theta}\right\|+\sup_{\theta\in\Theta}\left\|\frac{\partial\,\text{vec}\, b(\theta,Y_{-m-1})}{\partial\theta}\right\|\right),
%\end{align*}
%for some constant $\beta>0$. By ...., The sequence $\left(\frac{\partial\f[\theta]{\chunk{Y}{-k}{0}}(x)}{\partial\theta}\right)_{k\in\zsetp}$ is $\tilde\PP^\thv$-almost surely Cauchy and thus $\tilde\PP^\thv$-almost surely converges. It follows that this convergence is uniform and this implies, for all $x\in\Xset$ and $\theta\in\Theta$,
%\begin{align*}
%\lim_{m\to\infty}\frac{\partial\f[\theta]{\chunk{Y}{-m}{0}}(x)}{\partial\theta}=\frac{\partial}{\partial\theta}\lim_{m\to\infty}\f[\theta]{\chunk{Y}{-m}{0}}(x)=\frac{\partial\f[\theta]{\chunk{Y}{-\infty}{0}}}{\partial\theta},\; \tilde\PP^\thv\as,
%\end{align*}
%showing that the quantity $\frac{\partial\f[\theta]{\chunk{Y}{-\infty}{0}}}{\partial\theta}$ is well-defined $\tilde\PP^\thv$-almost surely.
%Similarly, we will next show that the sequence $\left(\frac{\partial^2\f[\theta]{\chunk{Y}{-k}{0}}(x)}{\partial\theta\partial \theta^T}\right)_{k\in\zsetp}$ is uniformly Cauchy $\tilde\PP^\thv$-almost surely by which we will show that




\subsection{Ergodicity}\label{sec:ergodicity:gen:od}




In this section, we provide conditions that yield stationarity and ergodicity of the Markov chain $\nsequence{(Z_k,Y_k)}{k\in\zsetp}$. We will set $\theta$ to be an arbitrary value in $\Theta$ and since this is a ``for all $\theta$
(...)'' condition, to save space and alleviate the notational burden, we will drop the
superscript $\theta$ from, for example, $G^\theta$, $R^\theta$ and
$\psi^\theta$ and respectively write $G$, $R$ and $\psi$, instead.

Ergodicity of Markov chains are usually studied using
$\varphi$-irreducibility. This approach is well known to be quite efficient when
dealing with fully dominated models; see \cite{meyn:tweedie:2009}. It is not at
all the same picture for integer-valued observation-driven models, where other tools need to
be invoked; see~\cite{fokianos:tjostheim:2011,dou:kou:mou:2013,douc2015handy}. Our result here is obtained in the same fashion as in \cite{douc2015handy}, which is inspired by the approach in \cite{dou:kou:mou:2013}. The following is the list of assumptions to obtain our main result on ergodicity. We should mention that some of our assumptions (for instance, Assumption \ref{assum:weakFeller-V:gen}) are more general than those derived in \cite{dou:kou:mou:2013} and \cite{douc2015handy}.

%Our main contribution here
%is to focus on an easy-to-check list of assumptions yielding the ergodicity
%conditions~\ref{assum:gen:identif:unique:pi:gen} and~\ref{ass:21-lyapunov:gen:od}. We also
%provide a lemma (\autoref{lem:det:alpha:gen:od}) which gives the construction of
%the instrumental functions $\alpha$ and $\phi$ used in the list of
%assumptions.

\begin{hyp}{AG}
\item \label{ass-ergo-Zset:gen} The measurable space $(\Zset, \Zmet)$ is a  locally
  compact, complete and separable metric space and its associated $\sigma$-field
  $\Zsigma$ is the Borel $\sigma$-field.

%\item \label{assum:weakFeller}
%The Markov kernel $R$ is weak Feller, that is, for any continuous and bounded
%function $f$ defined on $\Xset$, $Rf$ is continuous and bounded on $\Xset$.

\item \label{assum:weakFeller-V:gen} There exists a positive integer $q$ such that the Markov kernel $R^q$ is weak Feller, that is, for any continuous and bounded
function $f$ defined on $\Xset$, $R^qf$ is continuous and bounded on $\Xset$. Moreover, there exist $(\lambda,\beta) \in (0,1) \times
  \rsetp$ and a measurable function $V: \Zset \to \rsetp$ such that $R^q V \leq
  \lambda V+\beta$ and $\{V\le M\}$ is a compact set for any $M>0$.

\item \label{assum:reachable:gen}
The Markov kernel $R$ admits  a reachable point, that is, there exists
$z_0\in\Zset$ such that, for any $z\in\Zset$ and any neighborhood $\mathcal{N}$ of $z_0$,
$R^m(z;\mathcal{N})>0$ for at least one positive integer $m$.

% \item\label{assum:bound:rho:gen} There exist $(\rho,D)\in(0,1)\times\rsetp$ such that for all $n\in\zsetpnz$, $\chunk{y}{1}{n}\in\Yset^n$ and $(z,z')\in\Zset^2$ we have, $$\displaystyle{\Zmet(\F[ ]{\chunk{y}{1}{n}}(z),\F[ ]{\chunk{y}{1}{n}}(z'))}\le D\rho^{n}{\Zmet(z,z')}\eqsp.$$

 \item\label{assum:bound:rho:gen}  There exists a positive integer $\ell$ such that
\begin{equation}\label{eq:ineq:bound:rho:gen:ell}
\sup_{\substack{(z,z',\chunk{y}{1}{\ell})\in\Zset^2\times\Yset^\ell\\ z\neq z'}}\frac{\Zmet(\F[ ]{\chunk{y}{1}{\ell}}(z),\F[ ]{\chunk{y}{1}{\ell}}(z'))}{\Zmet(z,z')}<1\eqsp,
\end{equation}
and
\begin{equation}\label{eq:ineq:bound:rho:gen:1:k}
\sup_{\substack{(z,z',y)\in\Zset^2\times\Yset\\ z\neq z'}}\frac{\Zmet(\Psi_{y}(z),\Psi_{y}(z'))}{\Zmet(z,z')}<\infty\eqsp.
\end{equation}


 \item \label{assum:alpha-phi:gen} There exist a measurable function
   $\alpha$ from $\Zset^2$ to $[0,1]$, a measurable function
   $\phi:\Zset^2\to\Xset=\Proj{p}\Zset$ and a measurable function $W: \Zset^2 \to [1,\infty)$ such that the following assertions hold.
  \begin{enumerate}[label=(\roman*)]
  \item\label{item:alpha-phi:3:gen} For all $(z,z')\in\Zset^2$ and $y\in\Yset$,
\begin{equation}\label{eq:con:min-g-phi-gen:od}
\min\left\{g(\Proj{p}{z};y),g(\Proj{p}{z'};y)\right\}\ge\alpha(z,z')g\left(\phi(z,z');y\right)\eqsp.
\end{equation}
% \item\label{item:alpha:phi:equiv} For all $(x,x')\in\Xset^2$, $\alpha(x,x')=1$, then $G(x;\cdot)=G(x';\cdot)$, and then $G(x;\cdot)=G(\phi(x,x');\cdot)$.
\item  \label{assum:definition-gamma-x:gen} For all $z \in \Zset$, $W(z,\cdot)$ is finitely bounded in a
neighborhood of $z$, that is, there exists $\gamma_z>0$ such that
$\displaystyle\sup_{z' \in \ball{z}{\gamma_z}} W(z,z')<\infty$.
\item  \label{assum:hyp:1-alpha:W:gen}
For all $(z,z')\in\Zset^2$,
$1-\alpha(z,z')\leq \Zmet(z,z') W(z,z')$.
\item  \label{assum:driftCond:W:new:gen:od}
There exist an integer $m>0$ and a real number $D>0$ such that $$\displaystyle
 \sup_{(z,z')\in\Zset^2} \left( J_m((z,z');W) - W(z,z')\right)<\infty$$ and
 $$\sup_{(z,z')\in\Zset^2} \left( J_1((z,z');W) - DW(z,z')\right)<\infty,$$
where for any positive integer $n$, $(z,z')\in\Zset^2$ and measurable function $f\,:\,\Zset^2\to\rset$,
\begin{multline*}
J_n((z,z');f)\eqdef \int f\left(\F[]{\chunk{y}{1}{n}}(z),\F[]{\chunk{y}{1}{n}}(z')\right)\times \\
\prod_{k=1}^{n}G\left(\phi(\F[]{\chunk{y}{1}{k-1}}(z),\F[]{\chunk{y}{1}{k-1}}(z'));\rmd y_k\right)\eqsp.
\end{multline*}
  \end{enumerate}
\end{hyp}

\begin{remark}
Assumption~\ref{ass-ergo-Zset:gen} holds if the measurable metric spaces $(\Xset, \Xmet)$ and $(\Yset,\Ymet)$ are  locally compact, complete and separable and their associated $\sigma$-field
  $\Xsigma$ and $\Ysigma$ are the Borel $\sigma$-fields. In this case, the metric $\Zmet$ can be appropriately defined  as a function of $\Xmet$ and $\Ymet$.
\end{remark}

\begin{remark}\label{rem:weak:feller:genod}
Note that if the Markov kernel $R$ is weak Feller, then the Markov kernel $R^q$ is also weak Feller. %In many cases, it is sufficient to \ref{assum:weakFeller-V:gen} holds with $q=1$.
\end{remark}

\begin{lemma}
Assume \ref{ass-ergo-Zset:gen} and \ref{assum:weakFeller-V:gen}. Then $R$ admits an invariant probability distribution $\tilde\pi$; moreover, $\tilde\pi V<\infty$.
\end{lemma}
\begin{proof}
From Assumptions \ref{ass-ergo-Zset:gen} and \ref{assum:weakFeller-V:gen} and by \autoref{rem:weak:feller:genod}, the transition kernel $R^q$ admits an invariant probability distribution denoted by $\pi_q$. Let $\tilde\pi$ be defined by, for all $A\in\Zsigma$,
\begin{equation*}
\tilde\pi(A)=\frac{1}{q}\sum_{k=1}^q\pi_qR^k(A)\eqsp.
\end{equation*}
Obviously, we have $\tilde\pi R=\tilde\pi$, which shows that $R$ admits an invariant probability distribution $\tilde\pi$. Now let $M>0$. Then we have for all $n\in\zsetp$,
\begin{align*}
\tilde\pi  (V\wedge M)&=\tilde\pi R^{nq} (V\wedge M) \\
&\le \tilde\pi R^{nq} (V)\wedge M  \\
&\le\lambda^n\tilde\pi(V)\wedge M+\frac{\beta}{1-\lambda}\wedge M\eqsp.
\end{align*} Letting $n\to\infty$, we then obtain
\begin{equation*}
\tilde\pi  (V\wedge M)\le\frac{\beta}{1-\lambda}\wedge M.
\end{equation*}
Finally, since $M$ is arbitrary, by letting $M\to\infty$ and by monotone convergence theorem, we get $\tilde\pi V<\infty$ as desired.
\end{proof}

%\begin{remark}
%To check \ref{assum:bound:rho:gen} it is sufficient to show that there exists $\ell\in\zsetpnz$ such that
%\begin{equation}\label{eq:ineq:bound:rho:gen:ell}
%\sup_{\substack{(z,z',\chunk{y}{1}{\ell})\in\Zset^2\times\Yset^\ell\\ z\neq z'}}\frac{\Zmet(\F[ ]{\chunk{y}{1}{\ell}}(z),\F[ ]{\chunk{y}{1}{\ell}}(z'))}{\Zmet(z,z')}<1\eqsp,
%\end{equation}
%and
%\begin{equation}\label{eq:ineq:bound:rho:gen:1:k}
%\sup_{k\in\{1,\ldots,\ell-1\}}\sup_{\substack{(z,z',\chunk{y}{1}{k})\in\Zset^2\times\Yset^k\\ z\neq z'}}\frac{\Zmet(\F[ ]{\chunk{y}{1}{k}}(z),\F[ ]{\chunk{y}{1}{k}}(z'))}{\Zmet(z,z')}<\infty\eqsp.
%\end{equation}
%
%In many situations, Inequalities \eqref{eq:ineq:bound:rho:gen:ell} and \eqref{eq:ineq:bound:rho:gen:1:k} hold with $\ell=1$.
%\end{remark}

We can now state the main ergodicity result.

\begin{theorem}\label{thm:ergodicity:gen}
  Conditions~\ref{ass-ergo-Zset:gen}, \ref{assum:weakFeller-V:gen},  \ref{assum:reachable:gen}, \ref{assum:bound:rho:gen}
  and~\ref{assum:alpha-phi:gen} imply that $K$ admits a unique stationary
  distribution $\pi$ on $\Zset\times\Yset$. Moreover, $\pi_1 \bar V<\infty$ for
  every $\bar V:\Xset\to\rsetp$ such that $\bar V\lesssim V$.
\end{theorem}
For convenience, we postponed the proof of \autoref{thm:ergodicity:gen} to
\autoref{sec:ergodicity-proofs-gen:od}.
The first conclusion of \autoref{thm:ergodicity:gen} can directly be applied for
all $\theta\in\Theta$ to check~\ref{assum:gen:identif:unique:pi:gen}.
%The second
%conclusion can be used to check~\ref{ass:21-lyapunov:gen:od}. In doing so,
%one must take care of the fact that although $V$ may depend on $\theta$, $\bar
%V$ does not.

Assumptions~\ref{assum:weakFeller-V:gen} and~\ref{assum:reachable:gen}
have to be checked directly on the Markov kernel $R$ defined
by~\eqref{eq:def:kernel:R:gen}. To this end, it can be useful to define, for any given $z\in\Zset$, the
distribution
  \begin{equation}
    \label{eq:def-Pz:gen}
\bar\PP_z := \PP_{\delta_z\otimes G(\Proj{p}{z};\cdot)}
  \end{equation}
on $(\Zset\times\Yset)^{\zsetp}$,  where $\PP_\Xinit$ is
  defined for any distribution $\Xinit$ on $\Zset\times\Yset$ as in \autoref{def:equi:theta:gen:od}. Then the first component process
  $\{Z_k\,:\,k\in\zsetp\}$ associated to $\bar\PP_z$ is a Markov chain with Markov kernel $R$ and initial
  distribution $\delta_z$.

The following lemma provides a general way for constructing the instrumental functions $\alpha$ and $\phi$ that appear in~\ref{assum:alpha-phi:gen}. The proof can be easily adapted from \cite[Lemma~1]{douc2015handy} and is thus omitted.
\begin{lemma}\label{lem:det:alpha:gen:od}
  Suppose that $\Proj{p}{\Zset}=\Xset=\Cset^\Sset$ for some measurable space $(\Sset, \Ssigma)$
  and $\Cset\subseteq\rset$. Thus for all $x\in\Xset$, we write $x=(x_s)_{s\in
    \Sset}$, where $x_s\in\Cset$ for all $s\in\Sset$. Suppose moreover that for
  all $x=(x_s)_{s\in\Sset}\in\Xset$, we can express the conditional density
  $g(x;\cdot)$ as a mixture of densities of the form $j(x_s)h(x_s;\cdot)$ over
  $s\in\Sset$.  This means that for all $t\in\Cset$, $y\mapsto j(t)h(t;y)$ is a
  density with respect to $\nu$ and there exists a probability measure
  $\mu$ on $(\Sset, \Ssigma)$ such that
  \begin{equation}
    \label{eq:def-g-j-h-mu:gen:od}
g(x;y)=\int_\Sset {j(x_s)h(x_s;y)\mu(\rmd s)},\quad y\in\Yset\;.
\end{equation}
We moreover assume that $h$ takes nonnegative values and
that one of the two following assumptions holds.
\begin{hyp}{H'}
\item\label{item:F1:gen} For all $y\in\Yset$, the function $h(\cdot;y): t\mapsto h(t;y)$ is nondecreasing.
\item\label{item:F2:gen} For all $y\in\Yset$, the function $h(\cdot;y): t\mapsto h(t;y)$ is nonincreasing.
\end{hyp}
For all $(x,x')\in\Xset^2$, we denote $x\wedge x':=(\min\{x_s,x_s'\})_{s\in \Sset}$ and $x\vee
x':=(\max\{x_s,x_s'\})_{s\in \Sset}$ and we define $\alpha(z,z')$ and
$\phi(z,z')$ by: for all $(z,z')\in\Zset^2$, letting $x=\Proj{p}{z}$ and $x'=\Proj{p}{z'}$,
$$
\begin{cases}
\displaystyle\alpha(z,z')=\inf_{s\in \Sset}{\left\{\frac{j(x_s\vee x'_s)}{j(x_s\wedge
      x'_s)}\right\}} \quad\text{and}\quad\phi(z,z')=x\wedge x'
&\text{ under\emph{~\ref{item:F1:gen}}};\\
\displaystyle\alpha(z,z')=\inf_{s\in \Sset}{\left\{\frac{j(x_s\wedge x'_s)}{j(x_s\vee x'_s)}\right\}}
\quad\text{and}\quad\phi(z,z')=x\vee x'
&\text{ under\emph{~\ref{item:F2:gen}}}.
\end{cases}
$$
Then $\alpha$ and $\phi$ defined above
satisfy~\ref{assum:alpha-phi:gen}\ref{item:alpha-phi:3:gen}.% and~\ref{assum:alpha-phi:gen}\ref{item:alpha:phi:equiv}.
\end{lemma}
%\begin{proof}
%We only prove this result under Condition~\ref{item:F1:gen}. The proof is similar
%under~\ref{item:F2:gen}.
%
%Since for
%all $t\in\Cset$, $y\mapsto j(t)h(t;y)$ is a density with respect to $\nu$, we have
%\begin{equation*}
%j(t)=\left(\int{h(t;y)\nu(\rmd y)}\right)^{-1}>0\;.
%\end{equation*}
%Thus $j$ is non-increasing on $\Cset$.
%Clearly, the so-defined $\alpha$ takes values on $[0,1]$ and $\phi$
%defines a function from $\Zset^2$ to $\Proj{p}{\Zset}$.  For all $y\in\Yset$ and $(z, z')\in\Zset^2$ with $x=\Proj{p}{z}$ and $x'=\Proj{p}{z'}$, we have
%\begin{align*}
%g(x;y)&=\int_\Sset{j(x_s)h(x_s;y)\mu(\rmd s)} \\
%        &\ge \int_\Sset{j(x_s\vee x_s')h(x_s\wedge x_s';y)\mu(\rmd s)}\\
%        %&\ge \int{\frac{j(x_s)}{j(x_s\wedge x_s')}j(x_s\wedge x_s')h(x_s\wedge x_s';y)\mu(\rmd s)}\eqsp,\\
%        &\ge \int_\Sset{\frac{j(x_s\vee x_s')}{j(x_s\wedge x_s')}j(x_s\wedge x_s')h(x_s\wedge x_s';y)\mu(\rmd s)}\\
%        &\ge \int_\Sset{\inf_{s\in \Sset}\left\{\frac{j(x_s\vee x_s')}{j(x_s\wedge x_s')}\right\}j(x_s\wedge x_s')h(x_s\wedge x_s';y)\mu(\rmd s)}\\
%%        &\ge \alpha(x,x')\int_\Sset{j(x_s\wedge x_s')h(x_s\wedge x_s';y)\mu(\rmd s)}\\
%        &=\alpha(z,z')g(\phi(z,z');y)\eqsp.
%\end{align*}
%By symmetry of $\alpha$ and $\phi$, we get~\eqref{eq:con:min-g-phi-gen:od} and thus
%\ref{assum:alpha-phi:gen}\ref{item:alpha-phi:3:gen} holds.
%%Finally, to see that ~\ref{assum:alpha-phi:gen}\ref{item:alpha:phi:equiv} holds, let $(x,x')\in\Xset^2$ such that $\alpha(x,x')=1$. Then from the above result with the fact that $y\mapsto g(\cdot;y)$ is a probability density on $\Yset$, for all $y\mae$ in $\Yset$, we have
%%\begin{equation*}
%%g(x;\cdot)=g(x';\cdot)=g(\phi(x,x');\cdot)\eqsp,
%%\end{equation*}
%%which completes the proof.
%\end{proof}


%\section{Convergence of the MLE}
%\label{sec:cons-mle-od}


%\subsection{General result}
%Let us now introduce the
%following set of assumptions. The next four assumptions introduce additional
%restriction on the dependence of  $\psi^\theta_y(x)$ and $g^\theta(x;y)$ with
%respect to  $\theta$.
%\begin{hyp}{B}
%\item \label{assum:exist:x0:O} For all $\theta\neq\thv\in\Theta$, there exist $x \in\Xset$ and a
%  measurable function $\F[\theta,\thv]{\cdot}$ defined on
%  $\Yset^{\zsetn}$ such that
%\begin{equation}
%  \label{eq::exist:x0:O}
%\lim_{m \to \infty}\f{\chunk{Y}{-m}{0}}(x)=\F[\theta,\thv]{\chunk{Y}{-\infty}{0}}, \quad \tilde \PP^{\thv}\as
%\end{equation}
%\end{hyp}
%Suppose that
%Assumptions~\ref{assum:exist:x0:O}\ref{assum:continuity:Y:phi-theta:gen:od} hold and
%
%
%We also need the following condition on the convergence of the likelihood
%defined in~\eqref{eq:defi:lkdM:gen:od}.
%\begin{hyp}{B}
%\item \label{assum:limit:Y} Assumption~\ref{assum:exist:x0:O} holds, from
%  which, for all $\theta,\thv\in\Theta$, we define $p^{\theta,\thv}$
%  by~(\ref{eq:def-p1-od}). Then we suppose that there exists $x_1$ such that
%\begin{align}\label{eq:unfi-limit-contrast-od}
%%\lim_{k \to \infty} \sup_{\theta \in \Theta} \left|\ln \frac{g^\theta
%%(\f{\chunk{Y}{1}{k-1}}(x);Y_k)}{g^\theta
%%(\F[\theta,\thv]{\chunk{Y}{-\infty}{k-1}};Y_k)}\right|
%%=0\quad\tilde\PP^{\thv} \as
%\lim_{k \to \infty} \sup_{\theta \in \Theta} \left|\ln \frac{g^\theta (\f{\chunk{Y}{1}{k-1}}(x_1);Y_k)}{p^{\theta,\thv}(Y_k\,|\,\chunk{Y}{-\infty}{k-1})}\right| =0\quad\tilde\PP^{\thv} \as
%\end{align}
%\end{hyp}
%%Observe that the set of
%%conditions~\ref{assum:momentCond:gen:od}\ref{assum:limit:Y} implies
%%\ref{assum:exist:x0:O}\ref{assum:phi-cont:O}.
%%We can now state the consistency result which completes \autoref{thm:main:result:obs}.
%\begin{theorem}\label{thm:consis:O}
%Assume that~\ref{assum:gen:identif:unique:pi:gen} and~\ref{assum:momentCond:gen:od}\ref{assum:limit:Y} hold, then, we have
%\begin{equation}\label{eq:strong-consistency-prelim:gen:od}
%\lim_{n\to\infty}\met(\mlY{x_1,n},\Theta_\star)=0,\quad \tilde\PP^{\theta_\star}\as\;,
%\end{equation}
%where $x_1$ is the element of $\Xset$ appearing in~\ref{assum:limit:Y} and
%\begin{equation}
%\Theta_\star=\argmax_{\theta\in\Theta}\tilde{\PE}^{\thv}\left[\ln
%  p^{\theta,\thv}(Y_1|\chunk{Y}{-\infty}{0})\right]
%\end{equation}
%\end{theorem}



%\subsection{Practical conditions for consistency}
%\label{sec:practical-conditions-od-consistency}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%All the above listed conditions for the observation-driven models are easy to
%check except \ref{assum:gen:identif:unique:pi:gen} and~\ref{assum:limit:Y}
%(which also includes~\ref{assum:exist:x0:O}). \autoref{sec:exist:unic} is dedicated to the
%ergodicity of observation driven models and contains easy-to-check conditions
%to verify \ref{assum:gen:identif:unique:pi:gen}.  Here we provide practical
%conditions that imply~\ref{assum:limit:Y}.
%
%\begin{hyp}{P}
%\item \label{assum:practical-cond:1} Assumption~\ref{assum:practical-cond:O:gen:od}
%  hold and
%\begin{enumerate}[label=(\roman*)]
%\item \label{item:bar:phi}For all $\theta\in\Theta$, $\pi^\theta_2\left(\lnp (\bar{\phi})\right)<\infty$.
%\item \label{item:bar:phi:strong}  If $C>0$ then for all $\theta\in\Theta$, $\pi^\theta_2\left(\bar{\phi}\right)<\infty$.
%\end{enumerate}
%\end{hyp}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{lemma}\label{lem:exist:practical-cond:O}
%Assumptions~\ref{assum:practical-cond:O:gen:od},~\ref{assum:practical-cond:1} and~\ref{assum:continuity:Y:phi-theta:gen:od}
%imply~\ref{assum:limit:Y} with the same $x_1$ as in~\ref{assum:practical-cond:O:gen:od}.
%\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{proof}
%See \ref{app:lem}.
%\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




\section{Examples}\label{sec:examples:gen:od}
In this section, we apply our main results derived above to two models of interest, namely, the log-linear Poisson GARCH$(p,p)$ and the \acrshort{nbingarch}$(p,p)$  models. To the best of our knowledge, the stationarity and ergodicity as well as the asymptotic properties of the \gls{mle} for  the general log-linear Poisson GARCH$(p,p)$ and  \acrshort{nbingarch}$(p,p)$ models have not been derived so far.
In the sequel, $\mathbf{A}$, $\mathbf{b}$ and $\boldsymbol{\omega}$ always denote, respectively, the matrix
$$\left(\begin{array}{ccccccccccc}0 & 1 & 0 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0 & 0\\0 & 0 & 1 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0 & 0\\0 & 0 & 0 & 1 & \cdots & 0 & 0 & 0 & \cdots & 0 & 0\\\vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\0 & 0 & 0 & 0 & \cdots & 1 & 0 & 0 & \cdots & 0 & 0 \\a_p & a_{p-1} & a_{p-2} & a_{p-3} & \cdots & a_{1} & b_p & b_{p-1} & \cdots & b_3 & b_2 \\ 0 & 0 & 0 & 0 & \cdots & 0 & 0 &1 & \cdots & 0 & 0 \\0 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0 & 1 \\0 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0 & 0\end{array}\right)\eqsp,$$
 the vector $b_1\boldsymbol{\delta}_p+\boldsymbol{\delta}_{2p-1}$,
with for all $j\in\{1,\ldots,2p-1\}$, $\boldsymbol{\delta}_j=(\delta_{ij})_{i\in\{1,\ldots,2p-1\}}$, $\delta_{ij}$ being the Kronecker delta, and the vector $\omega\boldsymbol{\delta}_p$. We also denote by $\mathbf{A}_\star$, $\mathbf{b}_\star$ and $\boldsymbol{\omega}_\star$ the corresponding values of $\mathbf{A}$, $\mathbf{b}$ and $\boldsymbol{\omega}$ at $(\omega,\chunk{a}{1}{p},\chunk{b}{1}{p})=(\omega^\star,\chunk{a^\star}{1}{p},\chunk{b^\star}{1}{p})$, respectively. For any matrix $\mathbf{M}$, we denote its transpose by $\mathbf{M}^T$. For any real numbers $a$ and $b$, we denote their maximum value by $a\vee b$ or $\max\{a,b\}$ and their minimum value by $a\wedge b$ or $\min\{a,b\}$. \\

Now, let $\mathbf{I}_{n}$ be the identity matrix of order $n$ and denote the characteristic polynomial of the matrix $\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T$ by
$P(\lambda)=\det\left(\lambda\mathbf{I}_{2p-1}-(\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T)\right)$. For any square matrix $\mathbf{M}$, denote the
spectral radius of $\mathbf{M}$ by $\radius(\mathbf{M})$. Straightforward computation shows that
\begin{equation}\label{eq:P:char:poly:gen}
P(\lambda)=\lambda^{p-1}\left(\lambda^p-\sum_{k=1}^p(a_k+b_k)\lambda^{p-k}\right)\eqsp.
\end{equation}
%Since the matrix $\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T$ is nonnegative, the Perron-Frobenius theorem applies and we have the following result.
%\begin{lemma}\label{lem:perron-frobenius:nongetative}
%The spectral radius of $\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T$, $\radius(\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T)$, is nonnegative and the associated eigenvector $\mathbf{e}_{\max}$ has nonnegative entries.
%\end{lemma}
%If moreover $\chunk{a}{1}{p}$ and $\chunk{b}{1}{p}$ are all positive, a stronger result can be obtained according to the following proposition.
%
%\begin{proposition}\label{prop:per-frob:positive:primitive}
%If $\chunk{a}{1}{p}$ and $\chunk{b}{1}{p}$ are all positive, then $\radius(\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T)>0$ and the associated eigenvector $\mathbf{e}_{\max}$ has positive entries.
%\end{proposition}
%\begin{proof}
%By Perron-Frobenius theorem, it is sufficient to show that the matrix $\mathbf{B}=\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T$ is primitive, that is, there exists $k\in\zsetpnz$ such that all the entries of $\mathbf{B}^k$ is positive. Note that $\mathbf{B}$ is nonnegative, that is, all the entries of $\mathbf{B}$ is nonnegative. For simplicity, we will always assume \emph{ad hoc} that all the positive entries of $\mathbf{B}^t$, $t=1,2,\ldots$, are all equal to $1$, otherwise, equal to $0$. We will show that $\mathbf{B}^p$ has all entries equal to $1$ proving the lemma. First we compute $\mathbf{B}^2$. We have that all the entries of the $p$th row of $\mathbf{B}$ are equal to $1$. Since $\mathbf{B}_{p-1,p}$ and $\mathbf{B}_{2p-1,p}$, then the $p$th, $(p-1)$th and $(2p-1)$th of $\mathbf{B}^2$ contain all elements equal to $1$. Moreover, since $\mathbf{B}_{2p,2p-1}=1$, then $\mathbf{B}_{2p,p}^2=1$ as well. By continuing computing $\mathbf{B}^3$, $\mathbf{B}^4,\ldots$, with the same argument as above and by observing that after each calculation two new rows of all entries equal to $1$ are obtained, thus we have exactly in $p$ steps that all the entries of $\mathbf{B}^p$ are $1$ and therefore $\mathbf{B}$ is primitive.
%\end{proof}
The following lemma which is useful for the next example is immediate.
\begin{lemma}\label{lem:spec:rad:pos:1}
Let $\chunk{a}{1}{p}\ge 0$, $\chunk{b}{1}{p}\ge 0$. Then $\sum_{i=1}^p(a_i+b_i)<1$ if and only if
$$\radius(\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T)<1.$$
\end{lemma}

\begin{proof}
The proof is postponed to~\autoref{sec:proof:lem:spec:rad:pos:1}.
\end{proof}


\begin{remark}
It is well known that for any nonnegative matrices $\mathbf{M}=(m_{ij})$ and $\mathbf{N}=(n_{ij})$ such that $\mathbf{M}\ge\mathbf{N}$, that is for all $i, j$, $m_{ij}\ge n_{ij}$, we have $\radius(\mathbf{M})\ge\radius(\mathbf{N})$. Thus for our matrices $\mathbf{A}$ and $\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T$, we have $\radius(\mathbf{A})\le \radius(\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T)$.
%In fact this result can be shown directly by comparing the roots of the characteristic polynomials of $\mathbf{A}$ and of $\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T$. A direct calculation shows that the characteristic polynomial of $\mathbf{A}$ is $$P_{\mathbf{A}}(\lambda)=\lambda^{p-1}\left(\lambda^p-\sum_{i=1}^{p}a_i\lambda^{p-i}\right).$$
%Then we have $P(\lambda)=P_{\mathbf{A}}(\lambda)-\lambda^{p-1}\sum_{i=1}^{p}b_i\lambda^{p-i}$. If $\radius(\mathbf{A})=0$, then obviously $\radius(\mathbf{A})\le \radius(\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T)$. Otherwise, using intermediate value theorem one can show that $P(\lambda)$ has a root in $(\radius(\mathbf{A}),\infty)$.
\end{remark}



%Let us now apply these results to prove the convergence of MLE of Examples~\ref{example:Nbigarch:defi}, ~\ref{example:nmgarch:defi} and ~\ref{example:tingarch}.


%\input{nbigarch}
%\input{nmgarch}

%\begin{theorem}\label{theo:ergo-nmgarch}
%  Suppose that $\theta=\left(\boldsymbol{\gamma}, \boldsymbol{\omega}, \mathbf{A}, \mathbf{b}\right)\in\Theta$ satisfies Condition~\eqref{eq:con:station:nmgarch}.
%  Then $K^\theta$ has a  unique invariant probability measure
%  $\pi^\theta$. Moreover we have $\pi^\theta V <\infty$ with $\displaystyle V(z)=\|\mathbf{1}\|_\star\|z\|_\star$  for all $z\in\Xset$.
%\end{theorem}
%Assumption~\ref{assum:gen:identif:unique:pi:gen} holds and
%~\autoref{def:equi:theta:gen:od} makes sense if for all $\theta\in\Theta$, Condition~\eqref{eq:con:station:nmgarch} holds. Moreover, the consistency
%of the maximum likelihood estimator is obtained according to the following result.

%\begin{theorem}\label{theo:consistence-nmgarch}
%Let $\Theta$ be a compact subset of  $\simplex_d\times(0,\infty)^d\times\rsetp^{d\times d}\times\rsetp^d$ such that all
%$\theta=\left(\boldsymbol{\gamma}, \boldsymbol{\omega}, \mathbf{A}, \mathbf{b}\right)\in\Theta$ in $\Theta$ satisfies
%Condition~\eqref{eq:con:station:nmgarch}. Then for any $z\in\Xset$, the MLE
%$\mlY{z,n}$ defined by~\eqref{eq:defi:mle-od:gen} satisfies $\lim_{n \to \infty}
%\met(\mlY{z,n}, \Theta_\star) = 0$,\ $\tilde\PP^\thv\as$, where $\met$ is any distance on $\Theta$ %$\simplex_d\times(0,\infty)^d\times\rsetp^{d\times d}\times\rsetp^d$
% and $\Theta_\star$ ...\end{theorem}

%\subsection{GARCH$(p,p)$ Model}
%\begin{example}[Definition of \acrshort{garch}$(p,p)$]\label{exmpl:garch:gen} Consider \acrshort{garch}$(p,p)$, $p\in\zsetpnz$, defined as follows: for all $k\in\zsetp$,
%\begin{equation}\label{eq:def:garch:gen}
%\begin{split}
%&X_{k+1}=\omega+\sum_{i=1}^p{a_i}X_{k-i+1}+\sum_{i=1}^p{b_i}Y_{k-i+1}^2\eqsp, \\
%&Y_{k+1}\mid \mathcal{F}_k= X_{k+1}^{1/2}\varepsilon_{k+1}\eqsp,
%\end{split}
%\end{equation}
%where
%\begin{itemize}[label=--]
%\item
%the parameter space
%\begin{equation*}%\label{eq:def:theta:set}
%\Theta\subset\{\theta=(\omega,\chunk{a}{1}{p},\chunk{b}{1}{p})\in\rset\times\rset^{p}\times\rset^{p}\,:\, \omega>0, \chunk{a}{1}{p}\ge 0, \chunk{b}{1}{p}\ge 0\}\eqsp,
%\end{equation*} %and $\sum_{i=1}^p(a_i+b_i)> 0$
%\item
%$\mathcal{F}_k=\sigma\left(\chunk{X}{-p+1}{k+1},\chunk{Y}{-p+1}{k}\right)$,
%\item
%The process $\sequence{\varepsilon}[k][\zsetp]$ are {\iid} random sequence with zero mean and unit variance on $\rset$ and are independent of $\mathcal{F}_k$. For each $k\in\zsetp$, the distribution of $\varepsilon_{k}$ admits a density $\gamma$ \wrt \, the Lebesgue measure on $\rset$. The density $\gamma$ is assumed to be positive, to be continuous over $\rset$ and to satisfy that %\mo{Why we need to assume something about the density of $\varepsilon$? Will this assumption yield the ergodicity of the process?}
%$\gamma(u)+\gamma(-u)$ is monotone on $u\in\rsetp$. We further assume that $\epsilon>0$, $\PP(\varepsilon_{0}^2<\epsilon)>0$.
%\item
%$\Xset=(0,\infty)$, $\Xsigma=\mathcal{B}(\Xset)$, $\Yset=\rset$, $\Ysigma=\mathcal{B}(\Yset)$ and the metrics on $\Xset$ and $\Yset$ are any metrics on $\rset$.
%\end{itemize}
%
%Note that, by setting for all $k$, $U_k=Y_k^2$ (then $\Uset=\rsetp$, $\Usigma=\mathcal{B}(\Uset)$ and   $\Umet$ is any metric on $\rset$), Model~\eqref{eq:def:garch:gen} can be embedded in Model~\eqref{eq:def:garch:gen:trans} below:
%\begin{equation}\label{eq:def:garch:gen:trans}
%\begin{split}
%&X_{k+1}=\psi_{\chunk{U}{k-p+1}{k}}^\theta(\chunk{X}{k-p+1}{k})\eqsp, \\
%&U_{k+1}\mid \mathcal{F}_k= X_{k+1}\varepsilon_{k+1}^2\eqsp,
%\end{split}
%\end{equation}
%where for all $(\chunk{x}{1}{p},\chunk{u}{1}{p})\in\Xset^p\times\Uset^p$,
%\begin{equation}\label{def:psi:garch:gen:od}
%\psi_{\chunk{u}{k-p+1}{k}}^\theta(\chunk{x}{k-p+1}{k})\eqdef\omega+\sum_{i=1}^p{a_i}x_{p-i+1}+\sum_{i=1}^p{b_i}u_{p-i+1}.
%\end{equation}
%The density $g^\theta$ and the probability law $G^\theta$ of $U_k$ conditional on $X_k=x$ are respectively given by, for all $u\in\Uset$ and $A\in\Usigma$,
%\begin{equation}\label{eq:density:g:pos:gen}
%g(x;u)=\frac{1}{2\sqrt{xu}}\left[\gamma\left(\sqrt{x^{-1}u}\right)+\gamma\left(-\sqrt{x^{-1}u}\right)\right]>0\eqsp,
%\end{equation}
%and
%\begin{equation}\label{eq:density:G:pos:gen}
%G^\theta(x;A)=\int_{\Uset} \1_{A}(u)g^\theta(x;u)\rmd u\eqsp,
%\end{equation}
%with $g^\theta(x;0)=1$ for all $x\in\Xset$.
%In view of \autoref{def:obs-driv-gen}, Model~\eqref{eq:def:garch:gen:trans} is an \acrshort{odm}$(p,p)$ and is dominated by the Lebesgue measure on $\rset$. Now let $\Zset=\Xset^p\times\Uset^{p-1}$ and $\Zsigma=\Xsigma^{\otimes p}\otimes\Usigma^{\otimes {(p-1)}}$ and define the measurable function $\Psi^\theta$ similarly as in~\eqref{eq:def:psi:gen:od}.
%The function $\Psi^\theta$ then has the form: for all $(z,u)\in\Zset\times\Uset$,
%\begin{equation}\label{eq:psi:garch:gen}
%\Psi_{u}^\theta(z)=\boldsymbol{\omega}+\mathbf{A}z+u\mathbf{b}\eqsp.
%\end{equation}
%%where, for all $j\in\{1,\ldots,2p-1\}$, $\boldsymbol{\delta}_j=(\delta_{ij})_{i\in\{1,\ldots,2p-1\}}$, $\delta_{ij}$ is the Kronecker delta, $\boldsymbol{\omega}=\omega\boldsymbol{\delta}_p$,
%%$$\mathbf{A}=\left(\begin{array}{ccccccccccc}0 & 1 & 0 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0 & 0\\0 & 0 & 1 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0 & 0\\0 & 0 & 0 & 1 & \cdots & 0 & 0 & 0 & \cdots & 0 & 0\\\vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\0 & 0 & 0 & 0 & \cdots & 1 & 0 & 0 & \cdots & 0 & 0 \\a_p & a_{p-1} & a_{p-2} & a_{p-3} & \cdots & a_{1} & b_p & b_{p-1} & \cdots & b_3 & b_2 \\ 0 & 0 & 0 & 0 & \cdots & 0 & 0 &1 & \cdots & 0 & 0 \\0 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0 & 1 \\0 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0 & 0\end{array}\right)$$
%%and $\mathbf{b}=b_1\boldsymbol{\delta}_p+\boldsymbol{\delta}_{2p-1}$.
%Moreover, denoting $Z_k=(\chunk{X}{k-p+1}{k},\chunk{U}{k-p+1}{k-1})$ for all $k$, the process $\{Z_k: k\in\zsetp\}$ is a Markov chain on $(\Zset,\Zsigma)$ whose transition kernel $R^\theta$ is given by \eqref{eq:def:kernel:R:gen} with $G^\theta$ and $\Psi^\theta$ defined in \eqref{eq:density:G:pos:gen} and \eqref{eq:psi:garch:gen}, respectively.
%\end{example}
%
%Let $\mathbf{I}_{n}$ be the identity matrix of order $n$ and denote the characteristic polynomial of the matrix $\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T$ by
%$P(\lambda)=\det\left(\lambda\mathbf{I}_{2p-1}-(\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T)\right)$. For any square matrix $\mathbf{M}$, denote the
%spectral radius of $\mathbf{M}$ by $\radius(\mathbf{M})$. Straightforward computation shows that
%\begin{equation}\label{eq:P:char:poly:gen}
%P(\lambda)=\lambda^{p-1}\left(\lambda^p-\sum_{k=1}^p(a_k+b_k)\lambda^{p-k}\right)\eqsp.
%\end{equation}
%%Since the matrix $\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T$ is nonnegative, the Perron-Frobenius theorem applies and we have the following result.
%%\begin{lemma}\label{lem:perron-frobenius:nongetative}
%%The spectral radius of $\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T$, $\radius(\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T)$, is nonnegative and the associated eigenvector $\mathbf{e}_{\max}$ has nonnegative entries.
%%\end{lemma}
%%If moreover $\chunk{a}{1}{p}$ and $\chunk{b}{1}{p}$ are all positive, a stronger result can be obtained according to the following proposition.
%%
%%\begin{proposition}\label{prop:per-frob:positive:primitive}
%%If $\chunk{a}{1}{p}$ and $\chunk{b}{1}{p}$ are all positive, then $\radius(\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T)>0$ and the associated eigenvector $\mathbf{e}_{\max}$ has positive entries.
%%\end{proposition}
%%\begin{proof}
%%By Perron-Frobenius theorem, it is sufficient to show that the matrix $\mathbf{B}=\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T$ is primitive, that is, there exists $k\in\zsetpnz$ such that all the entries of $\mathbf{B}^k$ is positive. Note that $\mathbf{B}$ is nonnegative, that is, all the entries of $\mathbf{B}$ is nonnegative. For simplicity, we will always assume \emph{ad hoc} that all the positive entries of $\mathbf{B}^t$, $t=1,2,\ldots$, are all equal to $1$, otherwise, equal to $0$. We will show that $\mathbf{B}^p$ has all entries equal to $1$ proving the lemma. First we compute $\mathbf{B}^2$. We have that all the entries of the $p$th row of $\mathbf{B}$ are equal to $1$. Since $\mathbf{B}_{p-1,p}$ and $\mathbf{B}_{2p-1,p}$, then the $p$th, $(p-1)$th and $(2p-1)$th of $\mathbf{B}^2$ contain all elements equal to $1$. Moreover, since $\mathbf{B}_{2p,2p-1}=1$, then $\mathbf{B}_{2p,p}^2=1$ as well. By continuing computing $\mathbf{B}^3$, $\mathbf{B}^4,\ldots$, with the same argument as above and by observing that after each calculation two new rows of all entries equal to $1$ are obtained, thus we have exactly in $p$ steps that all the entries of $\mathbf{B}^p$ are $1$ and therefore $\mathbf{B}$ is primitive.
%%\end{proof}
%The following lemma is immediate.
%\begin{lemma}\label{lem:spec:rad:pos:1}
%Let $\chunk{a}{1}{p}\ge 0$, $\chunk{b}{1}{p}\ge 0$. Then $\sum_{i=1}^p(a_i+b_i)<1$ if and only if
%$$\radius(\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T)<1.$$
%\end{lemma}
%
%\begin{proof}
%The proof is postponed to~\autoref{sec:proof:lem:spec:rad:pos:1}.
%\end{proof}
%
%
%\begin{remark}
%It is well known that for any nonnegative matrices $\mathbf{M}=(m_{ij})$ and $\mathbf{N}=(n_{ij})$ such that $\mathbf{M}\ge\mathbf{N}$, that is for all $i, j$, $m_{ij}\ge n_{ij}$, we have $\radius(\mathbf{M})\ge\radius(\mathbf{N})$. Thus for our matrices $\mathbf{A}$ and $\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T$, we have $\radius(\mathbf{A})\le \radius(\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T)$.
%%In fact this result can be shown directly by comparing the roots of the characteristic polynomials of $\mathbf{A}$ and of $\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T$. A direct calculation shows that the characteristic polynomial of $\mathbf{A}$ is $$P_{\mathbf{A}}(\lambda)=\lambda^{p-1}\left(\lambda^p-\sum_{i=1}^{p}a_i\lambda^{p-i}\right).$$
%%Then we have $P(\lambda)=P_{\mathbf{A}}(\lambda)-\lambda^{p-1}\sum_{i=1}^{p}b_i\lambda^{p-i}$. If $\radius(\mathbf{A})=0$, then obviously $\radius(\mathbf{A})\le \radius(\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T)$. Otherwise, using intermediate value theorem one can show that $P(\lambda)$ has a root in $(\radius(\mathbf{A}),\infty)$.
%\end{remark}
%
% \begin{proposition}\label{theo:ergo-convergence:garch:gen}
%  Suppose that $\theta\in\Theta$ satisfies $\radius(\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T)<1$. Then the conclusion of \autoref{thm:ergodicity:gen} holds with $\bar V(z)=\mathbf{1}^Tz$ for all $z\in\Zset$.
%\end{proposition}
%
%\begin{remark}
%If $a_i\ge 0$, $b_i\ge 0$, for all $i\in\{1,\ldots,p\}$ and $\sum_{i=1}^p(a_i+b_i)<1$, then from \autoref{lem:spec:rad:pos:1}, \autoref{theo:ergo-convergence:garch:gen} holds with $\bar V(z)=\mathbf{1}^Tz$ for all $z\in\Zset$.
%\end{remark}
%%\begin{theorem}\label{theo:consistence-nmgarch}
%%Let $\Theta$ be a compact subset of  $\simplex_d\times(0,\infty)^d\times\rsetp^{d\times d}\times\rsetp^d$ such that all
%%$\theta=\left(\boldsymbol{\gamma}, \boldsymbol{\omega}, \mathbf{A}, \mathbf{b}\right)\in\Theta$ in $\Theta$ satisfies
%%Condition~\eqref{eq:con:station:nmgarch}. Then for any $z\in\Xset$ and any $\thv\in\Theta$, the MLE
%%$\mlY{z,n}$ defined by~\eqref{eq:defi:mle-od:gen} satisfies $\lim_{n \to \infty}
%%\met(\mlY{z,n}, [\thv]) = 0$,\ $\tilde\PP^\thv\as$, where $\met$ is any distance on $\simplex_d\times(0,\infty)^d\times\rsetp^{d\times d}\times\rsetp^d$ and $[\thv]$ is the equivalence class of $\theta_\star$ in the sense of
%%~\autoref{def:equi:theta:gen:od}.
%%\end{theorem}
%
%
%\begin{proof} Now set $\bar V(z)=V(z)=\mathbf{1}^Tz$, $z\in\Zset$. From \autoref{thm:ergodicity:gen}, we need to show that ~\ref{ass-ergo-Zset:gen}, \ref{assum:weakFeller-V:gen},  \ref{assum:reachable:gen}, \ref{assum:bound:rho:gen}
%  and~\ref{assum:alpha-phi:gen} hold. For convenience, we drop the superscript $\theta$ from for example $R^\theta$, $\Psi^\theta$, etc. Assumption~~\ref{ass-ergo-Zset:gen} holds with any metric $\Zmet$
%associated to a norm on the finite dimensional space $\Zset$. (The precise choice of this metric is
%postponed to the verification of~\ref{assum:bound:rho:gen}.)
%The fact that $R$ is weak Feller follows from the continuity of the density $\gamma$ and $\Proj{p}{\cdot}$. Moreover, we have, for all $z\in\Zset$,
%\begin{align*}
%RV(z)
%&=\int V(\boldsymbol{\omega}+\mathbf{A}z+u\mathbf{b})\;G(\Proj{p}{z};\rmd u)\\
%&=
%\mathbf{1}^T\boldsymbol{\omega}+\mathbf{1}^T\left(\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T\right)z\eqsp.
%\end{align*}
%By iteration, we obtain, for all $k\in\zsetpnz$,
%\begin{align*}
%R^kV(z)=\mathbf{1}^T\boldsymbol{\omega}+\cdots+\mathbf{1}^T\left(\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T\right)^{k-1}\boldsymbol{\omega}+\mathbf{1}^T\left(\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T\right)^kz\eqsp.
%\end{align*}
%Since $\radius(\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T)<1$, then there exist $(q,\rho)\in\zsetpnz\times(0,1)$ such that
%\begin{align*}
%R^qV(z)\le\mathbf{1}^T\boldsymbol{\omega}+\cdots+\mathbf{1}^T\left(\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T\right)^{q-1}\boldsymbol{\omega}+\rho^q\mathbf{1}^Tz\eqsp.
%\end{align*}
% Hence~\ref{assum:weakFeller-V:gen}
%holds with $\beta=\mathbf{1}^T\boldsymbol{\omega}+\cdots+\mathbf{1}^T\left(\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T\right)^{q-1}\boldsymbol{\omega}\ge0$ and $\lambda=\rho^q<1$. We now prove \ref{assum:reachable:gen}.
%We have for all $n\geq1$, $t_{0:n-1}\in\Uset^{n}$ and $z\in\Zset$,
%\begin{equation}\label{eq:psi:garch:iterate:gen:od}
%\F{\chunk{u}{0}{n-1}}(z)=
%\mathbf{A}^{n}z+\sum_{j=0}^{n-1}
%\mathbf{A}^j (\boldsymbol{\omega}+u_{n-1-j}\mathbf{b})\eqsp.
%\end{equation}
%Let us use the norm
%$$
%\|\mathbf{M}\|= \bigvee_{j=1}^{2p-1} \sum_{i=1}^{2p-1} |\mathbf{M}_{i,j}|=\sup_{|z|\leq1}|\mathbf{M}z|
%$$
%on $(2p-1)\times (2p-1)$ matrices, where $|z|=\sum_{i=1}^{2p-1}|z_i|$ is a norm of $z=\chunk{z}{1}{2p-1}\in\Zset$. Note that
%since $\radius(\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T)<1$, there exists $\delta\in(0,1)$ and $c>0$ such
%that, for any $k\geq1$,
%\begin{equation}
%  \label{eq:norm-control:gen:od}
%\left\|\left(\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T\right)^k\right\|\leq c\,
%\delta^k.
%\end{equation}
%Using that  $\mathbf{A}$, $\mathbf{b}$, $\boldsymbol{\delta}_p$ all have
%nonnegative entries, we have
%\begin{equation}
%  \label{eq:norm-control2:gen:od}
%\left\|\mathbf{A}^k\right\|\leq\left\|\left(\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T\right)^k\right\|.
%\end{equation}
%Hence $(\mathbf{I}_{2p-1}-\mathbf{A})^{-1}=\mathbf{I}_{2p-1}+\sum_{k\ge1}\mathbf{A}^k$ is well defined and we set
%$z_\infty=(\mathbf{I}_{2p-1}-\mathbf{A})^{-1}\boldsymbol{\omega}$ so that,
%with~(\ref{eq:psi:logpois:iterate:gen:od}), we have
%$$
%\F{\chunk{u}{0}{n-1}}(z)-z_\infty=
%\mathbf{A}^{n}z+ \sum_{j\geq n}\mathbf{A}^j \boldsymbol{\omega} +
%\sum_{j=0}^{n-1}
%u_{n-1-j}\mathbf{A}^j \mathbf{b}\;.
%$$
%Then, using definition~\eqref{eq:def-Pz:gen}, we get that, $\bar\PP_{z}$\as, for all $n\geq1$,
%\begin{align*}
%  |Z_n-z_\infty|&=\left|\F[]{\chunk{U}0{n-1}}(z)-z_\infty\right|\\
%&\leq
%\left|\mathbf{A}^n(z-z_\infty)\right|+ \sum_{j\geq n}\left|\mathbf{A}^j \boldsymbol{\omega}\right|+
%\left(\max_{0\leq j\leq n-1}U_j\right)\,\sum_{j=0}^{n-1}\left|\mathbf{A}^j\mathbf{b}\right|.
%\end{align*}
%With~(\ref{eq:norm-control:gen:od}) and~(\ref{eq:norm-control2:gen:od}), this implies
%$$
%\bar\PP_{z}\left(|Z_n-z_\infty|\leq c \left[\delta^n
%    \left(|z-z_\infty|+\frac{|\boldsymbol{\omega}|}{1-\delta}\right)+\frac{|\mathbf{b}|}{1-\delta}\,\max_{0\leq
%      j\leq n-1}U_j\right]\right) = 1.
%$$
%To obtain \ref{assum:reachable:gen}, it is sufficient to observe that, since $g$
%takes positive values in~\eqref{eq:density:g:pos:gen}, for any
%positive $\epsilon$, $z\in\Zset$ and any $n\geq1$,
%$$
%\bar\PP_{z}\left(\max_{0\leq j\leq n-1}U_j<\epsilon\right)>0.
%$$
%Next we prove~\ref{assum:bound:rho:gen}. We have
%$$
%\Psi_u(z)-\Psi_u(z')=\mathbf{A}(z-z').
%$$
%Since~\eqref{eq:norm-control:gen:od} and~\eqref{eq:norm-control2:gen:od} imply that
%$\radius(\mathbf{A})<1$, there exists a vector norm which makes $\mathbf{A}$
%strictly contracting. Choosing the metric $\Zmet$ on $\Zset$ (embedded in $\rset^{2p-1}$) as the one
%derived from this norm, we get~\ref{assum:bound:rho:gen}. To
%show~\ref{assum:alpha-phi:gen}, we rely
%on~\autoref{lem:det:alpha:gen:od}. Let us set $\Cset=(0,\infty)=\Xset$ and
%$\Sset=\{1\}$, then the probability measure $\mu$ boils down the Dirac mass on $\Sset$. For all $(x,u)\in\Xset\times\Uset$, we define:
%\begin{enumerate}[label=(\alph*)]
%\item\label{item:case:1:gen:od}
%if $\gamma(u)+\gamma(-u)$ is nonincreasing in $u\in\rsetp$,
%
%$\displaystyle j(x)=\sqrt{x^{-1}}$ and
%$h(x;u)=\frac{1}{2\sqrt{u}}\left[\gamma\left(\sqrt{x^{-1}u}\right)+\gamma\left(-\sqrt{x^{-1}u}\right)\right]$; and
%\item\label{item:case:2:gen:od}
%if $\gamma(u)+\gamma(-u)$ is nondecreasing in $u\in\rsetp$,
%
%$\displaystyle j(x)=\sqrt{x}$ and
%$h(x;u)=\frac{1}{2x\sqrt{u}}\left[\gamma\left(\sqrt{x^{-1}u}\right)+\gamma\left(-\sqrt{x^{-1}u}\right)\right]$.
%\end{enumerate}
%Clearly, \eqref{eq:def-g-j-h-mu:gen:od} holds. In Case~\ref{item:case:1:gen:od}, $h$
%satisfies ~\ref{item:F1:gen} and in Case~\ref{item:case:2:gen:od}, $h$ satisfies ~\ref{item:F2:gen}. Hence, by~\autoref{lem:det:alpha:gen:od}, we obtain
%$\alpha$ and $\phi$, which are given, respectively, as follows. For all $z,  z'\in\Zset$, in all cases, define
%$$\alpha(z,z')=\left(\frac{\Proj{p}{z}\wedge \Proj{p}{z'}}{\Proj{p}{z}\vee \Proj{p}{z'}}\right)^\frac{1}{2}$$
%and $\phi(z,z')=\Proj{p}{z}\wedge \Proj{p}{z'}$ in Case~\ref{item:case:1:gen:od} and $\phi(z,z')=\Proj{p}{z}\vee \Proj{p}{z'}$ in Case~\ref{item:case:2:gen:od}. These so-defined functions $\alpha$ and $\phi$ indeed satisfy ~\ref{assum:alpha-phi:gen}\ref{item:alpha-phi:3:gen}.  For any $z, z'\in\Zset$, we have
%\begin{align*}
%1-\alpha(z,z') &=1-\left(\frac{\Proj{p}{z}\wedge \Proj{p}{z'}}{\Proj{p}{z}\vee \Proj{p}{z'}}\right)^\frac{1}{2} \\
%&\leq  \Proj{p}{z}^{-1}\wedge\Proj{p}{z'}^{-1}\left|\Proj{p}{z}-\Proj{p}{z'}\right|\\
%& \leq W(z,z')\;\Zmet(z,z'),
%\end{align*}
%where $\Zmet$ is the metric previously defined and $W$ is defined by
%$W(z,z')=1\vee \left(c_{\Zmet} (\Proj{p}{z}^{-1}\wedge\Proj{p}{z'}^{-1})\right)$ with $c_{\Zmet}>0$ is
%conveniently chosen (such a constant exists since $\Zmet$ is the metric
%associated to a norm and $\Zset$ has finite dimension). Then
%~\ref{assum:alpha-phi:gen}\ref{assum:definition-gamma-x:gen} and
%~\ref{assum:alpha-phi:gen}\ref{assum:hyp:1-alpha:W:gen} hold and, since for all
%$u\in\Uset$ and $z\in\Zset$, $\Proj{p}{\Psi_u(z)}\ge\omega$,
%then $W(\Psi_u(z),\Psi_u(z'))$ is uniformly bounded over
%$(z,z', u)\in\Zset\times\Zset\times\Uset$
%and~\ref{assum:alpha-phi:gen}\ref{assum:driftCond:W:new:gen:od} holds. This
%completes the proof. %\textbf{Step 1}.
%\end{proof}

%\subsection{The Threshold INGARCH model}
%
%The threshold INGARCH$(1,1)$ in \autoref{example:tingarch} is a specific case of \autoref{def:obs-driv-gen} where $\nu$ is the counting measure on $\Yset=\zsetp$,
%\begin{align}
%&\psi^\theta_y(x)=\omega+ax+by\eqsp,\label{eq:tingarch:def:phi}\\
%&g^\theta(x;y)
%=\rme^{-(x\wedge \tau)} \frac{(x\wedge \tau)^y}{y!}\eqsp,\label{eq:tingarch:g}
%\end{align}
%with $\theta=(\omega, a, b, \tau)$ in a compact subset $\Theta$ of
%$(0,\infty)^4$ and $\Xset=(0,\infty)$. In this model, if $a<1$, we then have
%the ergodicity and consistency results as stated in
%\autoref{theo:ergo-converge:tingarch} below.
%
%\begin{theorem}\label{theo:ergo-converge:tingarch}
%  Suppose that all $\theta=(\omega, a, b, \tau)$ in $\Theta$ satisfy
%  $a<1$. Then
%  Assumptions~\ref{assum:gen:identif:unique:pi:gen}, \ref{ass:21-lyapunov:gen:od},
%  \ref{assum:continuity:Y:g-theta:gen:od}, \ref{assum:continuity:Y:phi-theta:gen:od} and \ref{assum:practical-cond:O:gen:od} hold
%  with $\bar V$ being defined as the identity function on $\Xset$ and with any $x_1\in\Xset$.
%\end{theorem}
%
%
%\begin{proof} As in the proofs of the two theorems above, for convenience, we
%  divide the proof into two steps.
%
%\noindent\textbf{Step 1.} We first prove
%Assumptions~\ref{assum:gen:identif:unique:pi:gen} and \ref{ass:21-lyapunov:gen:od} by
%applying \autoref{thm:ergodicity:gen}. We set $\bar V(x)= V(x)=x$ and thus we only need to
%check~~\ref{ass-ergo-Zset:gen}, \ref{assum:weakFeller-V:gen}, \ref{assum:weakFeller}, \ref{assum:reachable:gen},
%\ref{assum:bound:rho:gen} and~\ref{assum:alpha-phi:gen}. Condition~\ref{ass-ergo-Zset:gen} holds with the usual metric on $\rset$. We have for all $\theta\in\Theta$,
%$$
%R V(x)=\omega + ax+b(x\wedge\tau) \le aV(x) + (\omega+b\tau),
%$$
%which yields  \ref{assum:weakFeller-V:gen}.
%The fact that the kernel $R$ is weak Feller easily
%follows by observing that, as $x\to x'$, $\mathcal{P}(x)$ converges weakly
%to $\mathcal{P}(x')$ and the map $x\mapsto x\wedge\tau$ is continuous, so \ref{assum:weakFeller} holds.
%
%The proof of \ref{assum:reachable:gen} is similar to the NBIN-GARCH case of
%\autoref{theo:ergo-converge:nbigarch} and is thus omitted.
%% We now prove \ref{assum:reachable:gen}. Let $x_\infty=\omega/(1-a)$. Let $x \in
%% \rset$ and define recursively the sequence $x_0=x, x_k=\omega+ax_{k-1}$ for all
%% positive integers $k$. Since $0< a<1$, this sequence converges to the fixed
%% point $x_\infty$. Therefore, defining $\PP_x$ as in~(\ref{eq:def-Pz:gen}), for any
%% neighborhood $\mathcal{N}$ of $x_\infty$, there exists some $n$ such that
%% $x_n\in \mathcal{N}$ and we have
%% \begin{align*}
%% R^{n}(x; \mathcal{N})=\bar\PP_{x}\left(X_n\in
%%   \mathcal{N}\right)&\ge\bar\PP_{x}\left(X_k=x_k\text{ for all $k=1,\dots,n$}\right)\\
%% &=\bar\PP_{x}\left(Y_0=\ldots=Y_{n-1}=0\right)>0.
%% \end{align*}
%% So \ref{assum:reachable:gen} holds.
%Assumption~\ref{assum:bound:rho:gen} holds since we have for all
%$(x,x',y)\in\Xset^2\times\Yset$ with $x\neq
%x'$, $$\frac{|\psi_y(x)-\psi_y(x')|}{|x-x'|}=a<1\eqsp.
%$$
%To prove~\ref{assum:alpha-phi:gen}, we apply ~\autoref{lem:det:alpha:gen:od}
%with $\Cset=\Xset$, $\Sset=\{1\}$ (so $\mu$ boils down to the Dirac measure on $\{1\}$). For all
%$(x,y)\in\Xset\times\Yset$, let $j(x)=\rme^{-(x\wedge\tau)}$ and
%$h(x;y)=\frac{(x\wedge\tau)^y}{y!}$. Then $h$ indeed satisfies ~\ref{item:F1:gen}. Thus
%by~\autoref{lem:det:alpha:gen:od}, for all $(x,x')\in\Xset^2$ and $y\in\Yset$, we
%get that
%\begin{equation*}\label{eq:alpha:x:x'}
%\alpha(x, x')=\rme^{-(x\vee x')\wedge\tau+(x\wedge x')\wedge\tau}\in(0,1] \quad \text{and} \quad \phi(x,x')=x\wedge x'
%\end{equation*}
%satisfy ~\ref{assum:alpha-phi:gen}\ref{item:alpha-phi:3:gen}.
%
%Let  $W(x,x')=1$ for all $(x,x')\in\Xset^2$, which is a constant function. Thus \ref{assum:alpha-phi:gen}\ref{assum:definition-gamma-x:gen} and \ref{assum:alpha-phi:gen}\ref{assum:driftCond:W:new:gen:od}  clearly hold. Moreover, \ref{assum:alpha-phi:gen}\ref{assum:hyp:1-alpha:W:gen} holds since
%for all $(x, x') \in\Xset^2$, we have that
%\begin{equation*}
%1-\alpha(x,x')\leq x\vee x'-x\wedge x'=\lvert x-x'\rvert = W(x,x')\lvert x-x'\rvert\eqsp.
%\end{equation*}
%Therefore, \ref{assum:alpha-phi:gen} holds, which completes \textbf{Step 1}.
%
%\noindent\textbf{Step 2}. We now prove  %\ref{assum:momentCond:gen:od},
%  \ref{assum:continuity:Y:g-theta:gen:od}, \ref{assum:continuity:Y:phi-theta:gen:od} and \ref{assum:practical-cond:O:gen:od}.
%By assumption on $\Theta$, then there exists $(\underline{\omega},\bar{\omega},\underline{b}, \bar b,\underline{\tau}, \bar \tau,\underline{\alpha}, \bar \alpha) \in (0,\infty)^6\times(0,1)^2$ such that
%$$
%\underline{\omega}\leq \omega \leq\bar \omega, \,\, \leq b \leq\bar{b}, \,\, \underline{\tau}\leq \tau\leq \bar{r}, \,\, \underline{\alpha}\leq a\leq \bar{\alpha}\eqsp.
%$$
%Clearly, \ref{assum:continuity:Y:g-theta:gen:od} and
%\ref{assum:continuity:Y:phi-theta:gen:od} hold by definitions of $\psi^\theta_y(x)$
%and $g^\theta(x;y)$.  It remains to check~\ref{assum:practical-cond:O:gen:od} for a
%well-chosen closed subset $\Xset_1$ and any $x_1\in\Xset$. Let
%$\Xset_1=[\underline \omega,\infty)\subset\Xset$ so that
%\ref{assum:practical-cond:O:gen:od}\ref{assum:exist:practical-cond:subX:gen:od} holds. By
%noting that for all $(\theta,x,y)\in\Theta\times\Xset\times\Yset$,
%$g^\theta(x;y)\le1$, we have
%\ref{assum:practical-cond:O:gen:od}\ref{assum:momentCond:gen:od}.
%From~\eqref{eq:notationItere:f:cl} and ~\eqref{eq:tingarch:def:phi}, we have
%for all $s\leq t$, $y_{s:t}\in\Yset^{t-s+1}$, $x\in\Xset$ and
%$\theta\in\Theta$,
%\begin{equation}\label{eq:recurs:tingarch}
%\f{\chunk{y}{s}{t}}(x)= \omega\left(\frac{1-a^{t-s+1}}{1-a}\right)+a^{t-s+1}x+b\sum_{j=0}^{t-s} a^jy_{t-j} \eqsp.
%\end{equation}
%Using~\eqref{eq:recurs:tingarch}, we have, for all $\theta\in\Theta$, $x\in\Xset$ and $y_{1:n}\in\Yset^n$,
%\begin{equation*}
%\left|\f{\chunk{y}1n}(x_1)-\f{\chunk{y}1n}(x)\right|= a^n\left|x_1-x\right|\le\bar{\alpha}^n\left|x_1-x\right|\eqsp.
%  \end{equation*}
%  This gives~\ref{assum:practical-cond:O:gen:od}\ref{assum:exist:practical-cond:O:gen:od}
%  and~\ref{assum:practical-cond:O:gen:od}\ref{item:psibar:gen:od} by setting
%  $\varrho=\bar{\alpha}<1$ and $\bar\psi(x)=|x_1-x|$.  Next we set
%  $\bar\phi$, $H$ and $C$
%  to meet Conditions~\ref{assum:practical-cond:O:gen:od}\ref{item:barphi:bar:phi:gen:od}
%  and~\ref{assum:practical-cond:O:gen:od}\ref{item:ineq:loggg:gen:od} and~\ref{assum:practical-cond:O:gen:od}\ref{item:H}. Let us write, for all
%  $\theta\in\Theta$ and $y\in\Yset$,
%$$
%\left|x_1-\psi^\theta_y(x_1)\right|\le\omega+(1+a)x_1+by\le\bar\omega+(1+\bar{\alpha})x_1+\bar{b}y
%$$
%and, for all $(x,x')\in\Xset_1^2=[\underline \omega,\infty)^2$,
%\begin{align*}
%\left|\ln g^\theta (x;y)-\ln g^\theta (x';y)\right|
%&=\left|\left(x'\wedge\tau-x\wedge\tau\right)+y\left(\ln(x\wedge\tau)-\ln(x'\wedge\tau)\right)\right|\\
%&\leq \left(1+(\underline{\omega}\wedge\underline{\tau})^{-1}y\right)\;|x-x'| \;.
%\end{align*}
%Setting  $\bar\phi(y)=1+\bar\omega+(1+\bar{\alpha})x_1+\left(\bar{b}\vee(\underline{\omega}\wedge\underline{\tau})^{-1}\right)y$,
%$H(x)=x$ and $C=0$ then yield Conditions~\ref{assum:practical-cond:O:gen:od}\ref{item:barphi:bar:phi:gen:od}, ~\ref{assum:practical-cond:O:gen:od}\ref{item:ineq:loggg:gen:od}
%  and~\ref{assum:practical-cond:O:gen:od}\ref{item:H}. Now
%  \ref{assum:practical-cond:O:gen:od}\ref{item:22or23-barphi-V:gen:od} follows from
%$$
%\int \lnp y \; G^\theta(x,\rmd y) \leq \int y \; G^\theta(x,\rmd y) = x\wedge\tau\leq
%\bar V(x)\;.
%$$
%This concludes the proof.
%\end{proof}


%\section{Numerical experiments}
%\label{sec:numer-exper}
%\subsection{Numerical procedure}\label{sec:deriv:cal}
%In this part we provide a numerical method for computing the (conditional) MLE
%$\mlY{x,n}$ for the parameter $\theta=(\omega, a, b, r)$ in the
%NBIN-GARCH$(1,1)$ model introduced in \autoref{example:Nbigarch:defi} and
%studied in \autoref{sec:nbin-garch-model}. It is convenient to write
%$\theta=(\vartheta, r)$ with $\vartheta=(\omega, a, b)$ and then to write
%$\psi^{\vartheta}_y(x)$ and $g^r(x;y)$ instead of $\psi^{\theta}_y(x)$ and
%$g^\theta(x;y)$ in~(\ref{eq:Nbingarch:def:phi}) and~(\ref{eq:Nbingarch:g}),
%respectively. In contrast to the approach used in \cite{zhu:2011}, we allow the
%component $r$ to be any positive real number, rather than a discrete one and to
%be unknown as well. We thus maximize jointly with respect to the parameters
%$\vartheta$ and $r$ the log-likelihood function
%$\lkdM[x,n]{\chunk{y}{1}{n}}[\theta]=\lkdM[x,n]{\chunk{y}{1}{n}}[(\vartheta,
%r)]$. In practice, one does not rely on a compact set $\Theta$ of parameters
%as in~\autoref{theo:ergo-converge:nbigarch}. Instead the maximization is
%performed over all parameters $\omega>0$, $a>0$, $b>0$, $r>0$ such that  the stability
%constraint $a+br<1$ holds (taken from~(\ref{eq:nbingarch-stability-cond-zhu})).
%% . Mathematically, we solve the
%% constrained nonlinear optimization problem, with $\vartheta=(\omega, a, b)$ and
%% $\Theta$ being compact,
%% \begin{equation}\label{eq:prob:constr:log-like}
%% \begin{split}
%% \begin{aligned}
%% & \underset{(\vartheta, r)\in\Theta}{\text{maximize}}
%% & & \lkdM[x,n]{\chunk{y}{1}{n}}[(\vartheta, r)] \\
%% & \text{subject to}
%% & & a+br<1, \\
%% &&& {\omega>0,\,\, a\ge0,\,\, b\ge0,\,\, r>0}.
%% \end{aligned}
%% \end{split}
%% \end{equation}
%%\subsection{Estimation procedure}\label{sec:deriv:cal}
%%In this part we provide a (practical) method in estimating the (conditional) MLE $\mlY{x,n}$ for the parameter $\theta=(\omega, a, b, r)$ in the NBIN-GARCH$(1,1)$ model.  Though the component $r$ of $\theta$ can take any positive value, to make the analysis simple, we assume that $r$ takes values in a finite discrete set $F$. Now write $\theta=(\vartheta, r)$ with $\vartheta=(\omega, a, b)$. The procedure for estimating MLE $\mlY{x,n}$ consists of following steps (see for instance \cite{davis2009negative} and \cite{zhu:2011} in following this approach).
%%\begin{enumerate}
%%\item \label{step:max:vartheta} For a given value of $r$, compute
%%  $\hat\vartheta_r=(\hat\omega, \hat a, \hat b)$, the maximizer of
%%  $\lkdM[x,n]{\chunk{Y}{1}{n}}[(\vartheta,r)]$ which satisfies the constraints
%%  described in~\autoref{theo:ergo-converge:nbigarch}.
%%\item Compute $\hat r$, the maximizer of $\lkdM[x,n]{\chunk{Y}{1}{n}}[(\hat\vartheta_r,r)]$ on the finite set $F$.
%%\item Write $\mlY{x,n}=(\hat\vartheta_{\hat r}, \hat r)$.
%%\end{enumerate}
%% In our case, it may not be easy to obtain the closed form for the MLE
%% $\mlY{x,n}=(\hat\vartheta,\hat r)$, the maximizer of the problem
%% \eqref{eq:prob:constr:log-like} above. Thus the numerical method should be
%% used.
%We use the constrained nonlinear optimization function \texttt{auglag}
%(Augmented Lagrangian Minimization Algorithm) from the package \texttt{alabama}
%(Augmented Lagrangian Adaptive Barrier Minimization Algorithm) in \texttt{R}. For this purpose we provide an initial parameter
%point and a numerical computation of the normalized log-likelihood function
%$\lkdM[x,n]{\chunk{y}{1}{n}}[\theta]$ and of its gradient. The initial point is
%obtained  by applying a conditional least
%square (CLS) estimation based on an ARMA$(1,1)$ representation of the model, see \cite[Section~3]{zhu:2011}. The
%computation of the log-likelihood and of its
%derivatives are derived as follows. For all $x\in\Xset$, denoting
%$u_k^\vartheta=\F[\vartheta]{\chunk{y}{1}{k-1}} (x)$ for all $k\geq2$ and
%$u_1^\vartheta=x$, we have
%\begin{align*}
%%\label{eq:defi:lkdM:gen:od} \nonumber
%\lkdM[x,n]{\chunk{y}{1}{n}}[(\vartheta, r)]&= n^{-1} \sum_{k=1}^{n} \ln g^r\left(\F[\vartheta]{\chunk{y}{1}{k-1}} (x);y_k\right) \\
%&=n^{-1}\ln g^r(x,y_1)+ n^{-1}\sum_{k=2}^{n} \ln g^r\left(u_k^\vartheta;y_k\right) \eqsp.
%\end{align*}
%The computation of $u_k^\vartheta$ for all $k\geq2$ is done iteratively by
%observing that $u_k^\vartheta=\F[\vartheta]{y_{k-1}}(u_{k-1}^\vartheta)$ and the
%computation of $\lkdM[x,n]{\chunk{y}{1}{n}}[(\vartheta, r)]$ is deduced.
%The computation of the derivatives with respect to parameter $\theta=(\vartheta, r)$ of the
%function $\lkdM[x,n]{\chunk{y}{1}{n}}[(\vartheta, r)]$ are then obtained in two
%steps. First, for $k\geq2$, the derivative of $u_k^\vartheta$ with respect to $\vartheta$ are
%obtained iteratively by ${\partial{u_{1}^\vartheta}}/{\partial{\vartheta}}=0$ and
%\begin{equation*}
%\frac{\partial{u_k^\vartheta}}{\partial{\vartheta}}=
%(1, u_{k-1}^\vartheta, Y_{k-1})+a\frac{\partial{u_{k-1}^\vartheta}}{\partial{\vartheta}} \;.
%\end{equation*}
%Then the derivatives of $\lkdM[x,n]{\chunk{y}{1}{n}}[(\vartheta, r)]$ with respect to $\vartheta$ and $r$ are given by
%\begin{equation*}
%\begin{split}
%\frac{\partial{\lkdM[x,n]{}[(\vartheta,r)]}}{\partial{\vartheta}}&=n^{-1}\sum_{k=1}^n\frac{\partial{\ln g^r}}{\partial{x}}\left(u_k^\vartheta;y_k\right)\frac{\partial{u_k^\vartheta}}{\partial{\vartheta}}\\
%&=n^{-1}\sum_{k=2}^n\left(\frac{y_k}{u_k^\vartheta}-\frac{y_k+r}{1+u_k^\vartheta}\right)\frac{\partial{u_k^\vartheta}}{\partial{\vartheta}}
%\end{split}
%\end{equation*}
%and
%\begin{equation*}
%\begin{split}
%\frac{\partial{\lkdM[x,n]{}[(\vartheta,r)]}}{\partial{r}}&=n^{-1}\sum_{k=1}^n\frac{\partial{\ln g^r}}{\partial{r}}\left(u_k^\vartheta;y_k\right)\\
%&=n^{-1}\sum_{k=1}^n\left(\Gamma_2(r+y_k)-\ln(1+u_k^\vartheta)\right)-\Gamma_2(r)\eqsp,
%\end{split}
%\end{equation*}
%respectively, where $\Gamma_2$ is the digamma function $\Gamma_2(r)=\frac{\rmd}{\rmd{r}}{\ln\circ\Gamma(r)}$, $r>0$.

%Second derivatives with respect to $\vartheta$ are given by: for $k\ge 2$,
%\begin{equation*}
%\frac{\partial^2{u_k^\vartheta}}{\partial\vartheta^T\partial{\vartheta}}=
%\mathbf{Z}_{k-1}+\frac{\partial{u_{k-1}^\vartheta}}{\partial{\vartheta}}[0\,\,\,1\,\,\,0]+a\frac{\partial^2{u_{k-1}^\vartheta}}{\partial\vartheta^T\partial{\vartheta}} \,,
%\end{equation*}
%where
%\[
%\mathbf{Z}_{k-1}=
%  \begin{bmatrix}
%    0 & 0 & 0\\
%    {} & \left(\frac{\partial{u_{k-1}^\vartheta}}{\partial{\vartheta}}\right)^T & {} \\
%    0 & 0 & 0 \\
%  \end{bmatrix}\,.
%\]
%
%\begin{equation}
%\begin{split}
%\frac{\partial^2{\lkdM[x,n]{}[(\vartheta,r)]}}{\partial\vartheta^T\partial{\vartheta}}&=n^{-1}\sum_{k=2}^n\left[\frac{(Y_k+r)}{(1+u_k^\vartheta)^2}-\frac{Y_k}{(u_k^\vartheta)^2}\right]\frac{\partial{u_k^\vartheta}}{\partial{\vartheta}}\left(\frac{\partial{u_k^\vartheta}}{\partial{\vartheta}}\right)^T \\
%&+n^{-1}\sum_{k=2}^n\left[\frac{Y_k}{u_k^\vartheta}-\frac{(Y_k+r)}{1+u_k^\vartheta}\right]\frac{\partial^2{u_k^\vartheta}}{\partial\vartheta^T\partial{\vartheta}}\eqsp.
%\end{split}
%\end{equation}

%\subsection{Simulation study}
%We consider two NBIN-GARCH$(1,1)$ models with parameters:
%\begin{enumerate}
%\item[(M.1)] $\thv=(\omega_\star, a_\star, b_\star, r_\star)=(3,\,.2,\,.2,\,2)$ and
%\item[(M.2)] $\thv=(\omega_\star, a_\star, b_\star, r_\star)=(3,\,.35,\,.1,\,1.5)$.
%\end{enumerate}
%We simulated $m=200$ data sets for each sample size $n= 2^7, 2^8, 2^9$ and
%$2^{10}$.  In \autoref{pic-difflog}, we display the obtained boxplots of the
%difference of the normalized log-likelihood function evaluated respectively at
%MLE and at the true value $\thv$. As predicted by the theory, this difference
%appears to converge to $0$ as the number of observations $n\to\infty$. For the
%NBIN-GARCH$(1,1)$ model, it can be shown that $\Theta_\star=\{\thv\}$, which
%implies the convergence of the MLE to the true parameter. We can observe this
%behavior for each component of the MLE for the two models in
%\autoref{pic-mle:M1} and \autoref{pic-mle:M2}. We also report the Monte Carlo mean along with the mean
%absolute deviation error (MADE):
%$\text{MADE}=m^{-1}\sum_{j=1}^m|\hat\theta^j_{x,n}-\thv^j|$ as an evaluation
%criterion for the estimated parameter in
%\autoref{table:est}.
%
%% The result strongly suggests that as the sample size $n$
%% increases, the distance between the normalized log-likelihood function
%% evaluated at the MLE and at $\thv$, respectively, decreases. As shown in
%% \autoref{pic-mle:M1} and \autoref{pic-mle:M2} the biasedness of the MLE from
%% the true parameter is reduced with larger sample size $n$. Moreover, the result
%% also suggest that the estimated MLE does converge to the true parameter $\thv$
%% as $n$ tends to infinity.
%
%\begin{table}[h]
%\caption{\label{table:est}Mean of estimates, MADEs (within parentheses) for the NBIN-GARCH$(1,1)$ models}
%\begin{center}
%\begin{tabular}{r r r r r r l}
%\cline{1-6}					
% &  & \multicolumn{4}{c}{Sample size $n$} \\ \cline{3-6}
%Model & Parameter & $n=2^7$ & $n=2^8$ & $n=2^9$ & $n=2^{10}$ \\ \cline{1-6}
%\multicolumn{1}{c}{\multirow{4}{*}{(M.1)} } &
%\multicolumn{1}{c}{$\hat{\omega}$} & 3.311(.973) & 3.212(.719) & 3.108(.507) & 3.062(.372)      \\ %\cline{2-6}
%\multicolumn{1}{c}{}                        &
%\multicolumn{1}{c}{$\hat{a}$} & .165(.138) & .173(.113) & .187(.076) & .193(.055)     \\ %\cline{2-6}
%\multicolumn{1}{c}{}                        &
%\multicolumn{1}{c}{$\hat{b}$} & .194(.049) & .195(.034) & .197(.025) & .200(.018)      \\ %\cline{2-6}
%\multicolumn{1}{c}{}                        &
%\multicolumn{1}{c}{$\hat{r}$} & 2.045(.241) & 2.035(.166) & 2.020(.112) & 2.011(.074)      \\ \cline{1-6}
%
%\multicolumn{1}{c}{\multirow{4}{*}{(M.2)} } &
%\multicolumn{1}{c}{$\hat{\omega}$} & 3.525(1.325) & 3.362(1.258) & 3.326(1.041) & 3.167(.761)   \\ %\cline{2-6}
%\multicolumn{1}{c}{}                        &
%\multicolumn{1}{c}{$\hat{a}$} & .252(.227) & .290(.213) & .296(.170) & .319(.136)  \\ %\cline{2-6}
%\multicolumn{1}{c}{}                        &
%\multicolumn{1}{c}{$\hat{b}$} & .092(.056) & .097(.039) & .098(.028) & .100(.022)   \\ %\cline{2-6}
%\multicolumn{1}{c}{}                        &
%\multicolumn{1}{c}{$\hat{r}$} & 1.563(.175) & 1.539(.129) & 1.520(.093) & 1.513(.066)   \\ \cline{1-6}
%
%%\multicolumn{1}{ c  }{\multirow{4}{*}{B.3} } &
%%\multicolumn{1}{ c }{$\omega$} & 3 & 2 & 0 & 1 &   \\ %\cline{2-6}
%%\multicolumn{1}{ c  }{}                        &
%%\multicolumn{1}{ c }{$\hat{a}$} & 2 & 3 & 1 & 0 &   \\ %\cline{2-6}
%%\multicolumn{1}{ c  }{}                        &
%%\multicolumn{1}{ c }{$\hat{b}$} & 2 & 3 & 1 & 0 &     \\ %\cline{2-6}
%%\multicolumn{1}{ c  }{}                        &
%%\multicolumn{1}{ c }{$\hat{r}$} & 2 & 3 & 1 & 0 &     \\ \cline{1-6}
%\end{tabular}
%\end{center}
%\end{table}
%
%
%\begin{figure}[H]
%\centering
%%\textbf{The behavior of $\lkdM[x,n]{\chunk{y}{1}{n}}[\mlY{x,n}]-\lkdM[x,n]{\chunk{y}{1}{n}}[\thv]$ for Model (M.1) and (M.2)}
%%\includegraphics[scale=0.52, center]{M11.pdf}
%%\includegraphics[scale=0.52, center]{M21.pdf}
%\includegraphics[width=\textwidth,center]{M1M2.pdf}
%\caption[BehaviorDifflogM1:M2]{Boxplots of the differences of log-likelihood functions evaluated at the estimated MLE and the true value for Models (M.1) and (M.2) with sample sizes $n=2^7, 2^8, 2^9$ and $n=2^{10}$, respectively. The red ``continuous" line indicates the position of zero.}
%\label{pic-difflog}
%\end{figure}
%
%
%\begin{figure}[H]
%\centering
%%\textbf{The behavior of the MLE}
%\includegraphics[center]{M17.pdf}
%\caption[BhaviorMLE:M1]{Boxplots of the estimated MLE for Model (M.1) with
%  sample sizes $n=2^7, 2^8, 2^9$ and $n=2^{10}$, respectively. The red
%  ``dashed'' line indicates the true value of the parameter and the blue
%  ``\textsf{x}'' indicates the location of the Monte Carlo mean of the MLE.\label{pic-mle:M1}}
%\end{figure}
%
%\begin{figure}[H]
%\centering
%%\textbf{The behavior of the MLE}
%\includegraphics[center]{M27.pdf}
%\caption[BhaviorMLE:M2]{Same as \autoref{pic-mle:M1} but for Model (M.2).\label{pic-mle:M2}}
%\end{figure}
%
%
%
%% \subsection{Identifiability}
%% \mo{I think these conditions can yield $\Theta_\star=\{\thv\}$. Simulation study suggests the convergence of the MLE to $\thv$!}  Let $\theta=(\vartheta, r)$. Write $G^\theta=G^r$ and $\F[\theta]{\cdot}=\F[\vartheta]{\cdot}$ (that is, $G$ depends on $\theta$ explicitly only through $r$ and $\F[\theta]{\cdot}$ depends on $\theta$ only through $\vartheta$). This may work for NBIN-GARCH$(1,1)$. We have if  \rd{Reecrire a cet endroit}
%
%% \begin{enumerate}[(a).]
%% \item \label{item:ident:G:gen:od} $(r, x) \mapsto G^r(x;\cdot)$ is one-to-one, \ie, if  $G^r(x;\cdot) = G^{r'}(x',\cdot)$, then $r=r'$ and $x=x'$,
%% \item \label{item:ident:psi:gen:od}$\F[\vartheta]{\chunk{Y}{-\infty}{0}}=\F[\vartheta_\star]{\chunk{Y}{-\infty}{0}},\eqsp\PP^\thv\as$,  implies that $\vartheta=\vartheta_\star$,
%% \end{enumerate}
%% then $\Theta_\star=\{\thv\}$.
%
%% \begin{proof}
%% Let $\theta=(\vartheta, r)\in[\thv]$. Then  Kulback-Leibler divergence implies
%% \begin{equation*}
%% G^r(\F[\vartheta]{\chunk{Y}{-\infty}{0}};\cdot)=G^{r_\star}(\F[\vartheta_\star]{\chunk{Y}{-\infty}{0}};\cdot),\quad\tilde\PP^\thv\as
%% \end{equation*}
%% Then from \ref{item:ident:G:gen:od}, we obtain $r=r_\star$ and $\F[\vartheta]{\chunk{Y}{-\infty}{0}}=\F[\vartheta_\star]{\chunk{Y}{-\infty}{0}}$, $\tilde\PP^\thv\as$ and from \ref{item:ident:psi:gen:od}, we also have $\vartheta=\vartheta_\star$. Thus the proof follows.
%% \end{proof}
%
%
%
%% \begin{example}[NMGARCH(1,1)] \label{example:nmgarch:defi}
%% The model~:
%% \begin{equation}\label{eq:mmdm:def}
%% \begin{split}
%% &Z_{k+1}=\psi^\theta_{Y_k}(Z_k)\eqsp, \\
%% &Y_ {k+1}|\chunk{Z}{0}{k+1},\chunk{Y}{0}{k}\sim G^\theta(Z_{k+1};\cdot) \eqsp, \\
%% &g^\theta(Z_k;y)=\frac{\rmd G^\theta(Z_k;\cdot)}{\rmd\nu}(y)= \sum_{\ell=1}^{d}{\gamma_\ell\frac{\rme^{-y^2/{2X_{\ell,k}}}}{(2\pi X_{\ell,k})^{1/2}}}\eqsp, \quad y\in\rset\eqsp,
%% \end{split}
%% \end{equation}
%% where $\psi^\theta_{Y_k}(Z_k)=\boldsymbol{\omega}+\mathbf{A}Z_k+Y_{k}^2\mathbf{b}$, $\theta=(\boldsymbol{\gamma},\,\,\,\boldsymbol{\omega},\,\,\, \mathbf{A},\,\,\,\mathbf{b})$ and $\nu$ is the Lebesgue measure on $\rset$.
%% Now, let $\boldsymbol{\gamma}$ be given and hence one can considers $\theta=[\boldsymbol{\omega}\,\,\, \mathbf{A}\,\,\,\mathbf{b}]^T$ as a parameter.  In this case we will drop the superscript $\theta$ from $g^\theta$ and $\psi^\theta_{Y_k}(Z_k)$ can be expressed as:
%% \begin{equation}
%% \psi^\theta_{Y_k}(Z_k)=[\mathbf{I}\,\,\,Z_{k}^T\otimes\mathbf{I}\,\,\,Y_k^2\mathbf{I}]\theta
%% \end{equation}
%%  The normalized log likelihood function of the observations $\chunk{Y}{1}{n}$ conditionally on $Z_1=z_1$ is given by~:
%%  \begin{equation}
%% L^\theta_n=n^{-1}\ln p^\theta_{z_1}\left(\chunk{Y}{1}{n}\right)=n^{-1}\sum_{k=1}^n\ln g\left(u_k^\theta;Y_k\right)\eqsp,
%% \end{equation}
%% where
%% \begin{equation}
%% u_k^\theta = \begin{cases} z_1 &\mbox{ if } k = 1\,, \\
%% \psi_{Y_{k-1}}^\theta(u_{k-1}^\theta) & \mbox{ if } k > 1. \end{cases}
%% \end{equation}
%% \begin{enumerate}
%% \item{First derivative:} \\
%% For $k>1$,
%% \begin{equation}
%% \frac{\partial{u_k^\theta}}{\partial{\theta}}=\frac{\partial{\psi_{Y_{k-1}}^\theta}}{\partial{\theta}}(u_{k-1}^\theta)+\frac{\partial{\psi_{Y_{k-1}}^\theta}}{\partial{z}}(u_{k-1}^\theta)\frac{\partial{u_{k-1}^\theta}}{\partial{\theta}}\eqsp.
%% \end{equation}
%% Thus,
%% \begin{equation}
%% \frac{\partial{u_k^\theta}}{\partial{\theta}}=\begin{cases} \mathbf{0}_{d\times (d^2+2d)}  &\mbox{ if } k = 1\,, \\
%% [\mathbf{I}\,\,\,(u_{k-1}^\theta)^T\otimes\mathbf{I}\,\,\,Y_{k-1}^2\mathbf{I}]+\mathbf{A}\frac{\partial{u_{k-1}^\theta}}{\partial{\theta}} & \mbox{ if } k > 1\,. \end{cases}
%% \end{equation}
%% \begin{equation}
%% \begin{split}
%% \frac{\partial{L_n^\theta}}{\partial{\theta}}&=n^{-1}\sum_{k=1}^n \frac{1}{g\left(u_k^\theta;Y_k\right)}\frac{\partial{g}}{\partialz}\left(u_k^\theta;Y_k\right)\frac{\partial{u_k^\theta}}{\partial{\theta}}\\
%% &=n^{-1}\sum_{k=1}^n \frac{1}{(2\pi)^{1/2}g\left(u_k^\theta;Y_k\right)} \left[\frac{\gamma_{\ell}\left(Y_k^2- u_{k,\ell}^\theta\right)}{2(u_{k,\ell}^\theta)^{5/2}\rme^{{Y_k^2}/{2u_{k,\ell}^\theta}}}\right]_{\ell}\frac{\partial{u_k^\theta}}{\partial{\theta}}\eqsp.
%% \end{split}
%% \end{equation}
%
%% Note that for $d\times d$ matrix $A$ and a $d$-vector $x$
%% \begin{equation*}
%% A(x'\otimes I)=x'\otimes A
%% \end{equation*}
%
%%\item{Second derivative:}
%%\begin{equation}
%%\frac{\partial^2{u_k^\theta}}{\partial\theta^T\partial{\theta}}=\begin{cases} \mathbf{0}_{3\times 3}  &\mbox{ if } k = 1\,, \\
%%\mathbf{Z}_{k-1}+\frac{\partial{u_{k-1}^\theta}}{\partial{\theta}}(0\,\,\,1\,\,\,0)+a\frac{\partial^2{u_{k-1}^\theta}}{\partial\theta^T\partial{\theta}} & \mbox{ if } k > 1\,, \end{cases}
%%\end{equation}
%%where
%%\[
%%\mathbf{Z}_{k-1}=
%%  \begin{bmatrix}
%%    0 & 0 & 0\\
%%    {} & \left(\frac{\partial{u_{k-1}^\theta}}{\partial{\theta}}\right)^T & {} \\
%%    0 & 0 & 0 \\
%%  \end{bmatrix}\,.
%%\]
%%
%%\begin{equation}
%%\begin{split}
%%\frac{\partial^2{L_n^\theta}}{\partial\theta^T\partial{\theta}}&=n^{-1}\sum_{k=1}^n\left[\frac{(Y_k+r)}{(1+u_k^\theta)^2}-\frac{Y_k}{(u_k^\theta)^2}\right]\frac{\partial{u_k^\theta}}{\partial{\theta}}\left(\frac{\partial{u_k^\theta}}{\partial{\theta}}\right)^T \\
%%&+n^{-1}\sum_{k=1}^n\left[\frac{Y_k}{u_k^\theta}-\frac{(Y_k+r)}{1+u_k^\theta}\right]\frac{\partial^2{u_k^\theta}}{\partial\theta^T\partial{\theta}}\eqsp.
%%\end{split}
%%\end{equation}
%
%% \end{enumerate}
%
%% \end{example}
\subsection{Log-linear Poisson GARCH$(p,p)$ Model}
\begin{example}[Definition of the log linear Poisson GARCH$(p,p)$] Consider the log-linear Poisson GARCH of order $(p,p)$,  $p\in\zsetpnz$, which is defined as follows: for all $k\in\zsetp$,
\begin{equation}\label{eq:def:logpois:gen:od}
\begin{split}
&X_{k+1}=\omega+\sum_{i=1}^p{a_i}X_{k-i+1}+\sum_{i=1}^p{b_i}\ln(1+Y_{k-i+1})\eqsp, \\
&Y_{k+1}\mid \mathcal{F}_k\sim\mathcal{P}\left(\rme^{X_{k+1}}\right)\eqsp,
\end{split}
\end{equation}
where
\begin{itemize}[label=--]
\item
the parameter space
\begin{equation*}%\label{eq:def:theta:set:logpois:gen:od}
\theta=(\omega,\chunk{a}{1}{p},\chunk{b}{1}{p})\in\Theta\subset\rset^{2p+1}\eqsp,
\end{equation*}
\item
$\mathcal{F}_k=\sigma\left(\chunk{X}{-p+1}{k+1},\chunk{Y}{-p+1}{k}\right)$,
\item
$\Xset=\rset$, $\Xsigma=\mathcal{B}(\Xset)$, $\Yset=\zsetp$, $\Ysigma=\mathcal{B}(\Yset)$ and the metrics on $\Xset$ and $\Yset$ are the same usual metric on $\rset$.
\end{itemize}

By setting for all $k$, $U_k=\ln(1+Y_k)$ (hence $\Uset=\ln(1+\zsetp)$, $\Usigma=\mathcal{B}(\Uset)$ and $\Umet$ is the usual metric on $\rset$), then Model~\eqref{eq:def:logpois:gen:od} can be embedded in Model~\eqref{eq:def:logpois:gen:trans} below:
\begin{equation}\label{eq:def:logpois:gen:trans}
\begin{split}
&X_{k+1}=\psi_{\chunk{U}{k-p+1}{k}}^\theta(\chunk{X}{k-p+1}{k})\eqsp, \\
&U_{k+1}\mid \mathcal{F}_k\sim G^\theta(X_{k+1};\cdot)\eqsp,
\end{split}
\end{equation}
where for all $(\chunk{x}{1}{p},\chunk{u}{1}{p})\in\Xset^p\times\Uset^p$,
\begin{equation}\label{def:psi:logpois:gen:od}
\psi_{\chunk{u}{k-p+1}{k}}^\theta(\chunk{x}{k-p+1}{k})\eqdef\omega+\sum_{i=1}^p{a_i}x_{p-i+1}+\sum_{i=1}^p{b_i}u_{p-i+1}
\end{equation}
 and $G^\theta(x;\cdot)$ is the conditional law of $U_k$ given $X_k=x$ defined by
\begin{equation}\label{def:logpois:G:gen:od}
G^\theta(x;A)=\sum_{u\in A}\rme^{-\rme^x}\frac{\rme^{x(\rme^u-1)}}{(\rme^u-1)!}\eqsp, \quad A\in\Usigma.
\end{equation}
The density $g^\theta$ of $U_k$ conditionally on $X_k=x$ with respect to the counting measure on $\Uset$ is given by, for all $u\in\Uset$,
\begin{equation}\label{eq:density:g:logpois:gen:od}
g^\theta(x;u)=\rme^{-\rme^x}\frac{\rme^{x(\rme^u-1)}}{(\rme^u-1)!}>0\eqsp.
\end{equation}
By \autoref{def:obs-driv-gen}, Model~\eqref{eq:def:logpois:gen:trans} is an \acrshort{odm}$(p,p)$ and is dominated by the counting measure on $\Uset$. Now let $\Zset=\Xset^p\times\Uset^{p-1}$ and $\Zsigma=\Xsigma^{\otimes p}\otimes\Usigma^{\otimes {(p-1)}}$ and define the measurable function $\Psi^\theta$ similarly as in~\eqref{eq:def:psi:gen:od}.
The function $\Psi^\theta$ then take a simple linear form
\begin{equation}\label{eq:psi:logpois:gen:od}
\Psi_{u}^\theta(z)=\boldsymbol{\omega}+\mathbf{A}z+u\mathbf{b}\eqsp, \quad (z,u)\in\Zset\times\Uset\eqsp.
\end{equation}
%where, for all $j\in\{1,\ldots,2p-1\}$, $\boldsymbol{\delta}_j=(\delta_{ij})_{i\in\{1,\ldots,2p-1\}}$, $\delta_{ij}$ is the Kronecker delta, $\boldsymbol{\omega}=\omega\boldsymbol{\delta}_p$,
%$$\mathbf{A}=\left(\begin{array}{ccccccccccc}0 & 1 & 0 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0 & 0\\0 & 0 & 1 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0 & 0\\0 & 0 & 0 & 1 & \cdots & 0 & 0 & 0 & \cdots & 0 & 0\\\vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\0 & 0 & 0 & 0 & \cdots & 1 & 0 & 0 & \cdots & 0 & 0 \\a_p & a_{p-1} & a_{p-2} & a_{p-3} & \cdots & a_{1} & b_p & b_{p-1} & \cdots & b_3 & b_2 \\ 0 & 0 & 0 & 0 & \cdots & 0 & 0 &1 & \cdots & 0 & 0 \\0 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0 & 1 \\0 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0 & 0\end{array}\right)$$
%and $\mathbf{b}=b_1\boldsymbol{\delta}_p+\boldsymbol{\delta}_{2p-1}$.
Moreover, denoting $Z_k=(\chunk{X}{k-p+1}{k},\chunk{U}{k-p+1}{k-1})$ for all $k$, the process $\{Z_k: k\in\zsetp\}$ is a Markov chain on $(\Zset,\Zsigma)$ whose transition kernel $R^\theta$ is given by \eqref{eq:def:kernel:R:gen} with $G^\theta$ and $\Psi^\theta$ defined in \eqref{def:logpois:G:gen:od} and \eqref{eq:psi:logpois:gen:od}, respectively.
\end{example}

In this section, we will use the vector and the matrix norms defined respectively on $\rset^{2p-1}$ and the space of $(2p-1)\times (2p-1)$ matrices as follows:
$$
|z|_\infty=\bigvee_{i=1}^{2p-1}|\boldsymbol{\delta}_i^Tz|=\bigvee_{i=1}^{2p-1}|z_i|\eqsp,
$$
$$
\|\mathbf{M}\|_\infty= \bigvee_{i=1}^{2p-1} \sum_{j=1}^{2p-1} |\mathbf{M}_{i,j}|=\sup_{|z|_\infty\leq1}\left|\mathbf{M}z\right|_\infty.
$$
For notational convenience, we will also denote $\mathbf{A}_1=\mathbf{A}$ and $\mathbf{A}_2=\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T$.

 \begin{proposition}\label{theo:ergo-convergence:logpois:gen}
  Suppose that $\theta\in\Theta$ satisfies
  \begin{equation}\label{eq:def:par:set:logpois:gen}
 \bigvee_{(i_1,\ldots,i_p)\in\{1,2\}^p}\left\|\prod_{\ell=1}^p\mathbf{A}_{i_\ell}\right\|_\infty<1\eqsp.
  \end{equation}
 Then we have the following.
 \begin{enumerate}[label=(\roman*)]
 \item\label{item:thm:logpois:ergodic:gen:od}
\autoref{thm:ergodicity:gen} holds with the function $\bar V: \Zset\to\rsetp$ defined by,  for all $z\in\Zset$,
 \begin{equation}\label{eq:def:V:logpois:gen}
\bar V(z)=\rme^{\tau |z|_\infty}\eqsp,\quad \tau>0\eqsp.
 \end{equation}
 \item\label{item:thm:logpois:strong:consistency:gen:od}
\autoref{thm:convergence-main:gen:od} holds with any $z_1\in\Zset$. If the true parameter $\thv=(\omega^\star,\chunk{a^\star}{1}{p},\chunk{b^\star}{1}{p})$ moreover satisfies, for all $k\in\zsetpnz$,
\begin{equation}\label{eq:AA:star:b:gen:od}
(\mathbf{A}^k-\mathbf{A}_\star^k)\mathbf{b}_\star=0\quad \text{implies} \quad\mathbf{A}=\mathbf{A}_\star\eqsp,
\end{equation}
then \autoref{thm:strong:consistency:gen:od} also holds.
\item\label{item:thm:logpois:asymp:normal:gen:od}
If the true parameter $\thv$ lies in the interior of $\Theta$ and satisfies \eqref{eq:AA:star:b:gen:od},
%and, for all $k\in\zsetpnz$ and $\chunk{\alpha}{1}{2p-1}\in\rset^{2p-1}$, writing $\chunk{\theta}{1}{2p-1}=(\chunk{a}{1}{p},\chunk{b}{2}{p})$,
%\begin{align}\label{eq:invert:A:Astar:zero:gen:od}
%\sum_{i=1}^{2p-1}\alpha_i\boldsymbol{\delta}_p^T\left(\left.\frac{\partial\mathbf{A}^k}{\partial\theta_i}\right|_{\mathbf{A}=\mathbf{A}_\star}\right)\mathbf{b}_\star=0 \quad \implies \quad \chunk{\alpha}{1}{2p-1}=0\eqsp,
%\end{align}
%and
%\begin{align}\label{eq:invert:omega:zero:gen:od}
%\boldsymbol{\delta}_p^T(\mathbf{I}_{2p-1}-\mathbf{A}_\star)^{-1}\boldsymbol{\delta}_p\neq0\eqsp,
%\end{align}
then \autoref{thm:asym:normal:od:gen} holds.
 \end{enumerate}
\end{proposition}

\begin{remark}
When $p=1$,  then $\mathbf{A}_1=\mathbf{A}$ and $\mathbf{A}_2=\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T$ correspond to $a_1$ and $a_1+b_1$, respectively. Condition~\eqref{eq:def:par:set:logpois:gen} then reduces to
$$|a_1|\vee|a_1+b_1|<1.$$
This condition is weaker than the one derived in \cite{dou:kou:mou:2013} where $|b_1|<1$ is also imposed.
\end{remark}


\begin{remark}
To check \eqref{eq:AA:star:b:gen:od}, it suffices to check that there exists $k\in\zsetp$ such that the determinant
\begin{align*}
\det\left(\mathbf{A}_\star^k\mathbf{b}_\star,\mathbf{A}_\star^{k+1}\mathbf{b}_\star,\ldots,\mathbf{A}_\star^{k+2p-2}\mathbf{b}_\star\right)\neq0\eqsp.
\end{align*}
%is not identically zero, which is in turn equivalent to checking that the determinant
%\begin{align}\label{eq:det:conjecture:non:zero:gen:od}
%\left|\begin{array}{ccccc}
%\boldsymbol\delta_p^T\mathbf{b}_\star & \boldsymbol\delta_p^T\mathbf{A}_\star\mathbf{b}_\star & \cdots & \boldsymbol\delta_p^T\mathbf{A}_\star^{p-2}\mathbf{b}_\star & \boldsymbol\delta_p^T\mathbf{A}_\star^{p-1}\mathbf{b}_\star\\
%\boldsymbol\delta_p^T\mathbf{A}_\star\mathbf{b}_\star & \boldsymbol\delta_p^T\mathbf{A}_\star^2\mathbf{b}_\star & \cdots & \boldsymbol\delta_p^T\mathbf{A}_\star^{p-1}\mathbf{b}_\star & \boldsymbol\delta_p^T\mathbf{A}_\star^{p}\mathbf{b}_\star\\
%\vdots & \vdots & \ddots & \vdots & \vdots\\
%\boldsymbol\delta_p^T\mathbf{A}_\star^{p-2}\mathbf{b}_\star &\boldsymbol\delta_p^T\mathbf{A}_\star^{p-1}\mathbf{b}_\star & \ldots & \boldsymbol\delta_p^T\mathbf{A}_\star^{2p-4}\mathbf{b}_\star & \boldsymbol\delta_p^T\mathbf{A}_\star^{2p-3}\mathbf{b}_\star\\
%\boldsymbol\delta_p^T\mathbf{A}_\star^{p-1}\mathbf{b}_\star & \boldsymbol\delta_p^T\mathbf{A}_\star^{p}\mathbf{b}_\star & \ldots & \boldsymbol\delta_p^T\mathbf{A}_\star^{2p-3}\mathbf{b}_\star & \boldsymbol\delta_p^T\mathbf{A}_\star^{2p-2}\mathbf{b}_\star
%\end{array}\right|
%\end{align}
%does not identically vanish.
\end{remark}

\begin{remark}
When $p=1$, \eqref{eq:AA:star:b:gen:od} holds if and only if $b^\star\neq 0$.
%When $p=1$, \eqref{eq:AA:star:b:gen:od} and \eqref{eq:invert:A:Astar:zero:gen:od} hold if and only if $b^\star\neq 0$.
\end{remark}

Before proving \autoref{theo:ergo-convergence:logpois:gen}, let us show the following fact.
 \begin{lemma}\label{lem:support:bound:moment:pois}
 If $Y\sim\mathcal{P}(\zeta)$ with $\zeta>0$, then for any $\vartheta\in\rset$, there exist constants $c_1=c_1(\vartheta), c_2=c_2(\vartheta)>0$ such that
\begin{equation}\label{eq:lem:bound:moment:pois:gen1}
\PE[(1+Y)^\vartheta]\le c_1+c_2\zeta^\vartheta\1\{\vartheta>0\}.
\end{equation}
 \end{lemma}
 \begin{proof}
 The proof is postponed to~\autoref{sec:proof:lem:support:bound:moment:pois}.
 \end{proof}


\begin{proof} {\bf{Proof of \autoref{theo:ergo-convergence:logpois:gen}--\ref{item:thm:logpois:ergodic:gen:od}}}. Now set $V(z)=\bar V(z)=\rme^{\tau |z|_\infty}$, $z\in\Zset$. From \autoref{thm:ergodicity:gen}, we need to show that ~\ref{ass-ergo-Zset:gen}, \ref{assum:weakFeller-V:gen},  \ref{assum:reachable:gen}, \ref{assum:bound:rho:gen}
  and~\ref{assum:alpha-phi:gen} hold. Assumption~~\ref{ass-ergo-Zset:gen} holds with the metric $\Zmet$ defined by, for all $(z,z')\in\Zset^2$,
  \begin{equation}\label{eq:def:zmet:gen:od}
  \Zmet(z,z')=\bigvee_{i=1}^{2p-1}|z_i-z_i'|=|z-z'|_\infty\eqsp.
  \end{equation}
 The fact that $R$ is weak Feller follows from the continuity of the density $g$ with respect to the $x$ component in \eqref{eq:density:g:logpois:gen:od} and the continuity of $\Proj{p}{\cdot}$. We will show that, with $V$ defined in \eqref{eq:def:V:logpois:gen}, then we have $R^pV\le\lambda V+\beta$ for some $(\lambda,\beta)\in(0,1)\times\rsetp$. Note that \autoref{lem:support:bound:moment:pois} implies that, for all $\vartheta\in\rset$ and $\zeta>0$,
 \begin{align}\label{eq:ineq:moment:lp:gen:od}
 \int \rme^{\vartheta u}G\left(\zeta;\rmd u\right)\le c_1+c_2\rme^{\vartheta\zeta}\1\{\vartheta>0\},
 \end{align}
 where $c_1$ and $c_2$ are some positive constants.
Note further that for all $z\in\Zset$ and $\tau>0$, we have
 \begin{align}\label{eq:ineq:V:lp:gen:od}
 V(z)=\rme^{\tau|z|_\infty}\le\sum_{\sigma\in\{-1,1\}}\sum_{k=1}^{2p-1}\rme^{\sigma\tau z_k}\le 2(2p-1)\rme^{\tau|z|_\infty}
 \end{align}
 Without loss of generality, let us assume for the moment that $\tau=1$.
 Now, using \eqref{eq:ineq:moment:lp:gen:od} and \eqref{eq:ineq:V:lp:gen:od} and noting that, for all $z\in\Zset$, $z_k=\boldsymbol{\delta}_k^Tz$, $\Proj{p}{z}=\boldsymbol{\delta}_p^Tz$ and $\1\left\{\sigma\boldsymbol{\delta}_k^T\mathbf{b}>0\right\}\le1$, we have
 \begin{align*}
 RV(z)&\le\sum_{\sigma\in\{-1,1\}}\sum_{k=1}^{2p-1}\int \rme^{\sigma\boldsymbol{\delta}_k^T(\omega+\mathbf{A}z+u\mathbf{b})}G\left(\boldsymbol{\delta}_p^Tz;\rmd u\right)\\
 &\le\sum_{\sigma\in\{-1,1\}}\sum_{k=1}^{2p-1}\rme^{\sigma\boldsymbol{\delta}_k^T(\omega+\mathbf{A}z)}\int \rme^{\sigma\boldsymbol{\delta}_k^T\mathbf{b}u}G\left(\boldsymbol{\delta}_p^Tz;\rmd u\right) \\
 &\le\sum_{\sigma\in\{-1,1\}}\sum_{k=1}^{2p-1}\rme^{\sigma\boldsymbol{\delta}_k^T(\omega+\mathbf{A}z)}\left(\gamma_1+\gamma_2 \rme^{\sigma\boldsymbol{\delta}_k^T\mathbf{b}\boldsymbol{\delta}_p^Tz}\right)\eqsp,
 \end{align*}
 where $\gamma_1$ and $\gamma_2$ are some positive constants. This further yields that, with $\mathbf{A}_1=\mathbf{A}$ and $\mathbf{A}_2=\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T$,
  \begin{align*}
 RV(z) &\le\sum_{\sigma\in\{-1,1\}}\sum_{k=1}^{2p-1}\left(\beta_1\rme^{\sigma\boldsymbol{\delta}_k^T\mathbf{A}z}+\beta_2 \rme^{\sigma\boldsymbol{\delta}_k^T(\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T)z}\right)\eqsp\\
& \le\sum_{\sigma\in\{-1,1\}}\sum_{k=1}^{2p-1}\left(\beta_1\rme^{\sigma\boldsymbol{\delta}_k^T\mathbf{A}_1z}+\beta_2 \rme^{\sigma\boldsymbol{\delta}_k^T\mathbf{A}_2z}\right),
 \end{align*}
where $\beta_1$ and $\beta_2$ are some positive constants.
 By iteration, $R^{m+1}V=R(R^mV)$, we obtain that, for all $m\in\zsetpnz$,

  \begin{align}\label{eq:ineq:iterate:RV:lp:gen:od}
 R^mV(z) & \le\sum_{\sigma\in\{-1,1\}}\sum_{k=1}^{2p-1}\sum_{(i_1,\ldots,i_p)\in\{1,2\}^m}\beta_{i_1,\ldots,i_m}\rme^{\sigma\boldsymbol{\delta}_k^T\left(\prod_{\ell=1}^m\mathbf{A}_{i_\ell}\right)z} \\ \nonumber
 & \le\sum_{(i_1,\ldots,i_m)\in\{1,2\}^m}\beta_{i_1,\ldots,i_m}\sum_{\sigma\in\{-1,1\}}\sum_{k=1}^{2p-1}\rme^{\sigma\boldsymbol{\delta}_k^T\left(\prod_{\ell=1}^m\mathbf{A}_{i_\ell}\right)z}  \\ \nonumber
  & \le\sum_{(i_1,\ldots,i_m)\in\{1,2\}^m}\beta_{i_1,\ldots,i_m}\sum_{\sigma\in\{-1,1\}}\sum_{k=1}^{2p-1}\rme^{\left\|\prod_{\ell=1}^p\mathbf{A}_{i_\ell}\right\|_\infty|z|_\infty} \\ \nonumber
   & \le2(2p-1)\sum_{(i_1,\ldots,i_m)\in\{1,2\}^m}\beta_{i_1,\ldots,i_m}\rme^{\left\|\prod_{\ell=1}^m\mathbf{A}_{i_\ell}\right\|_\infty|z|_\infty} \\ \nonumber
   & \le\left(2(2p-1)\sum_{i_1,\ldots,i_m\in\{1,2\}}\beta_{i_1,\ldots,i_m}\right)\rme^{\bigvee_{i_1,\ldots,i_m\in\{1,2\}}\left\|\prod_{\ell=1}^m\mathbf{A}_{i_\ell}\right\|_\infty|z|_\infty},
 \end{align}
 where, for all $(i_1,\ldots,i_m)\in\{1,2\}^m$, $\beta_{i_1,\ldots,i_m}$ are positive constants.
Thus, for all $z\in\Zset$, with $m=p$, we get
$$
R^pV(z)\le \gamma_p\rme^{\eta_p|z|_\infty},
$$
where
$$\gamma_p\eqdef2(2p-1)\sum_{(i_1,\ldots,i_p)\in\{1,2\}^p}\beta_{i_1,\ldots,i_p}>0$$
and
$$\eta_p\eqdef\bigvee_{(i_1,\ldots,i_p)\in\{1,2\}^p}\left\|\prod_{\ell=1}^p\mathbf{A}_{i_\ell}\right\|_\infty<1.$$
Since
\begin{equation*}
\lim_{|z|_\infty\to\infty}\frac{\mathbf{e}^{\eta_p|z|_\infty}}{V(z)}=0\eqsp
\end{equation*}
and for all $M>0$,
\begin{equation*}
\sup_{|z|_\infty\le M}R^pV(z)<\infty\eqsp,
\end{equation*}
then there exists a pair $(\lambda,\beta)\in(0,1)\times\rsetp$ such that $R^pV\le\lambda V+\beta$, and hence~\ref{assum:weakFeller-V:gen}
holds. Note that when $m=p-1$ and $\tau=1$, we have $\rme^{|\Proj{p}{z}|}\le \gamma_{p-1} R^{p-1}V(z)$ for some $\gamma_{p-1}>0$. We now turn to prove \ref{assum:reachable:gen}.
Note that
from~\eqref{eq:def:par:set:logpois:gen}, there exist $\delta\in(0,1)$ and $c>0$ such
that, for any $k\geq1$,
\begin{equation}
  \label{eq:norm-control:logpois:gen:od}
\left\|\mathbf{A}^k\right\|_\infty\leq c\,\delta^k.
\end{equation}
Hence, $(\mathbf{I}_{2p-1}-\mathbf{A})^{-1}=\mathbf{I}_{2p-1}+\sum_{k\ge1}\mathbf{A}^k$ is well defined. Now, set
$z_\infty=(\mathbf{I}_{2p-1}-\mathbf{A})^{-1}\boldsymbol{\omega}$. Let $\mathcal{N}_\infty$ be an open neighborhood of $z_\infty$ and let $z\in\Zset$. Define recursively the sequence $\sequence{z}[k][\zsetp]$ with $z_0 = z$ and for all $k\ge1$, $z_k=\boldsymbol{\omega}+\mathbf{A}z_{k-1}$. From~\eqref{eq:norm-control:logpois:gen:od}, this so-defined sequence admits a unique limiting point $\lim_{n\to\infty}z_n=z_\infty$. Thus, there exists some $n\in\zsetpnz$ such that for all $k\ge n$, $z_k\in\mathcal{N}_\infty$. For such $n$, we have
\begin{align*}
R^n(z;\mathcal{N}_\infty)=\bar\PP_{z}(Z_n\in\mathcal{N}_\infty)&\ge\bar\PP_{z}(Z_k=z_k \,\, \text{for all} \,\, k=1,\ldots,n)\\
&=\bar\PP_{z}(U_0=\cdots=U_{n-1}=0)>0\eqsp.
\end{align*}

Next we prove~\ref{assum:bound:rho:gen}. We have for all $n\geq1$, $u_{0:n-1}\in\Uset^{n}$ and $z\in\Zset$,
\begin{equation}\label{eq:psi:logpois:iterate:gen:od}
\F[]{\chunk{u}{0}{n-1}}(z)=
\mathbf{A}^{n}z+\sum_{j=0}^{n-1}
\mathbf{A}^j (\boldsymbol{\omega}+u_{n-1-j}\mathbf{b})\eqsp.
\end{equation}
Then for all $(z,z')\in\Zset^2$, $\chunk{u}{0}{p-1}\in\Uset^p$,
\begin{align*}
\Zmet(\Psi_{u_0}(z),\Psi_{u_0}(z'))&=|\mathbf{A}(z-z')|_\infty \\
&\le \|\mathbf{A}\|_\infty|z-z'|_\infty\\
&\le D\Zmet(z,z')\eqsp,
\end{align*}
and
\begin{align*}
\Zmet(\F[]{\chunk{u}{0}{p-1}}(z),\F[]{\chunk{u}{0}{p-1}}(z'))&=|\mathbf{A}^p(z-z')|_\infty \\
&\le \|\mathbf{A}^p\|_\infty|z-z'|_\infty\\
&\le \rho\Zmet(z,z')\eqsp,
\end{align*}
where $D=\|\mathbf{A}\|_\infty>0$ and $\rho=\|\mathbf{A}^p\|_\infty<1$ from~\eqref{eq:def:par:set:logpois:gen}, showing ~\ref{assum:bound:rho:gen}.
 To show~\ref{assum:alpha-phi:gen}, we rely
on~\autoref{lem:det:alpha:gen:od}. Let us set $\Cset=\rset=\Xset$ and
$\Sset=\{1\}$, then the probability measure $\mu$ boils down the Dirac mass on $\Sset$. For all $(x,u)\in\Xset\times\Uset$, let $j(x)=\rme^{-\rme^x}$ and
$h(x;u)=\frac{\rme^{x(\rme^u-1)}}{(\rme^u-1)!}$. Indeed,
$h$ satisfies ~\ref{item:F1:gen}. Hence, from~\autoref{lem:det:alpha:gen:od}, the instrumental functions
$\alpha$ and $\phi$ are obtained by setting: for all $z,  z'\in\Zset$,
$$\alpha(z,z')=\frac{\rme^{-\rme^{\Pi_{p}(z)\vee\Pi_{p}(z')}}}{\rme^{-\rme^{\Pi_{p}(z)\wedge\Pi_{p}(z')}}}
=\rme^{-\left|\rme^{\Pi_p(z)}-\rme^{\Pi_p(z')}\right|}
$$
and $\phi(z,z')=\Proj{p}{z}\wedge \Proj{p}{z'}$. These so-defined functions $\alpha$ and $\phi$ indeed satisfy ~\ref{assum:alpha-phi:gen}\ref{item:alpha-phi:3:gen}.  For $(z, z')\in\Zset^2$, we have
\begin{align*}
1-\alpha(z,z') &=1-\rme^{-\left|\rme^{\Pi_p(z)}-\rme^{\Pi_p(z')}\right|}\le \left|\rme^{\Pi_p(z)}-\rme^{\Pi_p(z')}\right|\\
&\leq  \rme^{|\Proj{p}{z}|\vee|\Proj{p}{z'}|}\left|\Proj{p}{z}-\Proj{p}{z'}\right|\\
& \leq W(z,z')\;\Zmet(z,z'),
\end{align*}
 where $W$ is defined by, for all $z,z'\in\Zset$,
$$W(z,z')\eqdef \gamma_{p-1}\left(R^{p-1}V_1(z)+R^{p-1}V_1(z')\right)$$
and $V_1(z)=\rme^{|z|_\infty}$. Obviously, this so-defined function $W$ satisfies ~\ref{assum:alpha-phi:gen}\ref{assum:definition-gamma-x:gen} and
~\ref{assum:alpha-phi:gen}\ref{assum:hyp:1-alpha:W:gen}. To complete the proof of \autoref{theo:ergo-convergence:logpois:gen}--\ref{item:thm:logpois:ergodic:gen:od}, it remains to show ~\ref{assum:alpha-phi:gen}\ref{assum:driftCond:W:new:gen:od}. Note that \ref{assum:weakFeller-V:gen} holds with $V(z)=\rme^{\tau|z|_\infty}$ for any $\tau>0$, thus it also holds with $V_1$.  For all $z,z'\in\Zset$,
\begin{align*}
&J_1((z,z');W)= \int W(\Psi_u(z),\Psi_u(z'))
G\left(\phi(z,z');\rmd u\right) \\
=&\gamma_{p-1}\int R^{p-1}V_1(\Psi_u(z))
G\left(\phi(z,z');\rmd u\right)+\gamma_{p-1}\int V_1(\Psi_u(z'))
G(\phi(z,z');\rmd u).
\end{align*}
From \eqref{eq:ineq:iterate:RV:lp:gen:od}, with $m=p-1$, we have
\begin{multline}\label{eq:ineq:trans:RV:gen:od}
\int R^{p-1}V_1(\Psi_u(z))G\left(\phi(z,z');\rmd u\right) \\
\le\sum_{\sigma\in\{-1,1\}}\sum_{k=1}^{2p-1}\sum_{(i_1,\ldots,i_{p-1})\in\{1,2\}^{p-1}}\beta_{i_1,\ldots,i_{p-1}}\times \\
\int\rme^{\sigma\boldsymbol{\delta}_k^T\left(\prod_{\ell=1}^{p-1}\mathbf{A}_{i_\ell}\right)(\omega+\mathbf{A}z+u\mathbf{b})}G\left(\phi(z,z');\rmd u\right).
\end{multline}
Note that by \eqref{eq:ineq:moment:lp:gen:od}, there exist constants $c'_1$ and $c'_2$ such that
\begin{align*}
&\int\rme^{\sigma\boldsymbol{\delta}_k^T\left(\prod_{\ell=1}^{p-1}\mathbf{A}_{i_\ell}\right)\mathbf{b}u}G\left(\phi(z,z');\rmd u\right) \\
&\le c'_1+c'_2\rme^{\sigma\boldsymbol{\delta}_k^T\left(\prod_{\ell=1}^{p-1}\mathbf{A}_{i_\ell}\right)\mathbf{b}\left(\boldsymbol{\delta}_p^Tz\wedge \boldsymbol{\delta}_p^Tz'\right)}\1\left\{\sigma\boldsymbol{\delta}_k^T\left(\prod_{\ell=1}^{p-1}\mathbf{A}_{i_\ell}\right)\mathbf{b}>0\right\} \\
&\le c'_1+c'_2\rme^{\sigma\boldsymbol{\delta}_k^T\left(\prod_{\ell=1}^{p-1}\mathbf{A}_{i_\ell}\right)\mathbf{b}\boldsymbol{\delta}_p^Tz},
\end{align*}
where we have used $\boldsymbol{\delta}_p^Tz\wedge \boldsymbol{\delta}_p^Tz'\le\boldsymbol{\delta}_p^Tz$ and $\1\left\{\sigma\boldsymbol{\delta}_k^T\left(\prod_{\ell=1}^{p-1}\mathbf{A}_{i_\ell}\right)\mathbf{b}>0\right\}\le1$. Plugging this inequality into \eqref{eq:ineq:trans:RV:gen:od} and noting that $\mathbf{A}_1=\mathbf{A}$ and $\mathbf{A}_2=\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T$, then there exist positive constants $\gamma_{i_1,\ldots,i_{p}}$, $(i_1,\ldots,i_{p})\in\{1,2\}^{p}$, such that
\begin{align*}
\int R^{p-1}V_1(\Psi_u(z))&G\left(\phi(z,z');\rmd u\right) \\
&\le\sum_{\sigma\in\{-1,1\}}\sum_{k=1}^{2p-1}\sum_{(i_1,\ldots,i_{p})\in\{1,2\}^{p}}\gamma_{i_1,\ldots,i_{p}}\rme^{\sigma\boldsymbol{\delta}_k^T\left(\prod_{\ell=1}^{p}\mathbf{A}_{i_\ell}\right)z} \\
&\le \left(2(2p-1)\sum_{(i_1,\ldots,i_{p})\in\{1,2\}^{p}}\gamma_{i_1,\ldots,i_{p}}\right)\rme^{\eta_p|z|_\infty},
\end{align*}
where $\eta_p=\bigvee_{(i_1,\ldots,i_p)\in\{1,2\}^p}\left\|\prod_{\ell=1}^p\mathbf{A}_{i_\ell}\right\|_\infty<1$. Similarly, we have
\begin{align*}
\int R^{p-1}V_1(\Psi_u(z'))G\left(\phi(z,z');\rmd u\right) \le \left(2(2p-1)\sum_{i_1,\ldots,i_{p}\in\{1,2\}}\gamma'_{i_1,\ldots,i_{p}}\right)\rme^{\eta_p|z'|_\infty}
\end{align*}
for some positive constants $\gamma'_{i_1,\ldots,i_{p}}$, $(i_1,\ldots,i_{p})\in\{1,2\}^{p}$.
Thus, there exists  $M>0$ such that for all $z,z'\in\Zset$,
$$
J_1((z,z');W)=M\left(\rme^{\eta_p|z|_\infty}+\rme^{\eta_p|z'|_\infty}\right).
$$
Assumption \ref{assum:alpha-phi:gen}\ref{assum:driftCond:W:new:gen:od} follows by observing that
\begin{equation*}
\lim_{|z|_\infty\vee|z'|_\infty\to\infty}\frac{J_1((z,z');W)}{W(z,z')}=0\eqsp.
\end{equation*}
This completes the proof of \autoref{theo:ergo-convergence:logpois:gen}-- \ref{item:thm:logpois:ergodic:gen:od}. \smallskip \\

{\bf{Proof of \autoref{theo:ergo-convergence:logpois:gen}--\ref{item:thm:logpois:strong:consistency:gen:od}}}. As a consequence, we have ~\ref{assum:gen:identif:unique:pi:gen}. Assumption~\ref{ass:21-lyapunov:gen:od} directly follows.
Thus, to show the first part of \autoref{theo:ergo-convergence:logpois:gen}--\ref{item:thm:logpois:strong:consistency:gen:od}, we only need to show that Assumptions \ref{assum:continuity:Y:g-theta:gen:od}, \ref{assum:continuity:Y:phi-theta:gen:od} and \ref{assum:practical-cond:O:gen:od} are satisfied. The second part then follows if we can moreover show that \ref{item:ident:G:gen:od} and \ref{item:ident:psi:gen:od} in \autoref{thm:strong:consistency:gen:od} hold.

Clearly, \ref{assum:continuity:Y:g-theta:gen:od} and
  \ref{assum:continuity:Y:phi-theta:gen:od} hold by the definitions of
  $\psi^\theta$ and $g^\theta$ given by \eqref{def:psi:logpois:gen:od} and \eqref{eq:density:g:logpois:gen:od}, respectively. It remains to show
  ~\ref{assum:practical-cond:O:gen:od}. Since $\Theta$ is compact, then
\begin{equation}\label{eq:set:bound:lpois:consist:gen:od}
 |\omega|\leq\overline{\omega}, \,
|a_i| \leq\bar a,\,
 |b_i| \leq\bar b,\,
 \bigvee_{{(i_1,\ldots,i_p)}\in\{1,2\}^p}\left\|\prod_{\ell=1}^p\mathbf{A}_{i_\ell}\right\|_\infty\le\bar{\rho}, \left\|\mathbf{A}\right\|_\infty\le L,
\end{equation}
for some $(\overline{\omega},\bar a,\overline{b},\bar \rho) \in (0,\infty)^3\times(0,1)$ and $L>0$.
%By \cite[Lemma~12]{moulines-priouret-roueff-2005}, we note that this implies
%that, for all $\bar\delta\in(\bar\rho,1)$, there exists $\bar C>0$ such that for all
%$k\geq1$ and all $\theta\in\Theta$,
%\begin{equation}
%  \label{eq:uniform-spectral-raius:logpois:gen:od}
%  \left\|\left(\mathbf{A}+\mathbf{b}\boldsymbol{\gamma}^T\right)^k\right\|\leq
%  \bar C\,\bar\delta^k\;.
%\end{equation}
We set
$\Zset_1=\Zset$. And so
\ref{assum:practical-cond:O:gen:od}\ref{assum:exist:practical-cond:subX:gen:od}
holds. Moreover, for all
$(\theta,z,u)\in\Theta\times\Zset_1\times\Uset$,
$g^\theta(\Proj{p}{z};y)\le 1$. Thus,
Condition~\ref{assum:practical-cond:O:gen:od}\ref{assum:momentCond:gen:od} holds.  Now let
$z_1\in\Zset$.
Using ~\eqref{eq:norm-control:logpois:gen:od}, \eqref{eq:psi:logpois:iterate:gen:od} and \eqref{eq:set:bound:lpois:consist:gen:od}, we get that there exist constants $\bar c>0$ and $\bar\delta\in(0,1)$ such that for all $z\in\Zset$, $u_{1:n}\in\Uset^n$ and $\theta\in\Theta$,
\begin{align*}
\left|\F{\chunk{u}{1}{n}}(z_1)-\F{\chunk{u}{1}{n}}(z)\right|_\infty&= \left|\mathbf{A}^n(z_1-z)\right|_\infty\\
&\leq \bar c\,\bar\delta^n\left|z_1-z\right|_\infty\eqsp.
  \end{align*}
Thus by \eqref{eq:def:zmet:gen:od}, we get~\ref{assum:practical-cond:O:gen:od}\ref{assum:exist:practical-cond:O:gen:od} with
$$
\bar\Psi(z)= c\,\left|z_1-z\right|_\infty\eqsp.
$$
Hence~\ref{assum:practical-cond:O:gen:od}\ref{item:psibar:gen:od} holds and since
$$
\left|z_1-\Psi^\theta_u(z_1)\right|\le (L+1)\,|z_1|_\infty+\overline{\omega}+(1\vee\bar b)u,
$$
we also get~\ref{assum:practical-cond:O:gen:od}\ref{item:barphi:bar:phi:gen:od} provided that
\begin{equation}
  \label{eq:phibar-logpois-temp:gen:od}
\bar\phi(u)\ge (L+1)\,|z_1|_\infty+\overline{\omega}+(1\vee\bar b)u.
\end{equation}
It is straightforward to show that for all $\theta\in\Theta$, $(x,x')\in\Proj{p}{\Zset_1}\times\Proj{p}{\Zset_1}$ and $u\in\Uset$,
\begin{align*}
\left|\ln g^\theta\left(x;u\right)-\ln g^\theta\left(x';u\right)\right|&\le |x-x'|\rme^{|x|\vee|x'|}\rme^u \\
&\le|x-x'|\rme^{\left|x-\Proj{p}{z_1}\right|\vee\left|x'-\Proj{p}{z_1}\right|}\rme^{u+\left|\Proj{p}{z_1}\right|}.
\end{align*}
Thus  for all $\theta\in\Theta$, $(z,z')\in\Zset_1\times\Zset_1$ and $u\in\Uset$,
\begin{align*}
\left|\ln g^\theta\left(x;u\right)-\ln g^\theta\left(x';u\right)\right|\le|z-z'|_\infty\rme^{|z-z_1|_\infty\vee|z'-z_1|_\infty}\rme^{u+\left|\Proj{p}{z_1}\right|}.
\end{align*}
We thus obtain~\ref{assum:practical-cond:O:gen:od}\ref{item:barphi:bar:phi:gen:od},
\ref{assum:practical-cond:O:gen:od}\ref{item:ineq:loggg:gen:od}
and~\ref{assum:practical-cond:O:gen:od}\ref{item:H:gen:od} by setting $C=1$,
$$
H(s)=s,\quad s\in\rsetp,
$$
and
$$
\bar\phi(u)=(L+1)\,|z_1|_\infty+\overline{\omega}+(1\vee\bar b)u+\rme^{u+\left|\Proj{p}{z_1}\right|}.
$$
In addition, for all $\theta\in\Theta$, $x\in\Xset$ and $\tau>0$, we have
$$
\int \rme^{\tau u} G^\theta(x;\rmd u) \le 1+2^\tau\rme^{\tau x} \;.
$$
Hence, by letting $\bar V=V$ and using that for all $\theta\in\Theta$, $\pi_1^\theta V<\infty$, with the above definitions, we
obtain~\ref{assum:practical-cond:O:gen:od}\ref{item:22or23-barphi-V:gen:od}. This establishes the equivalence-class consistency of the \gls{mle} for the log-linear Poisson GARCH$(p,p)$.


To conclude the strong consistency of the \gls{mle}, it remains to check \ref{item:ident:G:gen:od} and \ref{item:ident:psi:gen:od} in \autoref{thm:strong:consistency:gen:od}. Condition~\ref{item:ident:G:gen:od} clearly holds by the identifiability of the Poisson distribution. Note that as a consequence of the earlier proof, we have, for all $\theta\in\Theta$, $\tilde\PP^\thv\as$,
\begin{equation*}%\label{eq:def:psi:stationary:derive:gen:od}
\F[\theta]{\chunk{U}{-\infty}{0}}=
\sum_{k=0}^{\infty}
\mathbf{A}^k (\boldsymbol{\omega}+U_{-k}\mathbf{b})\eqsp.
\end{equation*}
By stationarity, for all $\theta\in\Theta$ and $t\in\zset$, we have, $\tilde\PP^\thv\as$,
\begin{equation*}%\label{eq:def:psi:stationary:derive:gen:od}
\F[\theta]{\chunk{U}{-\infty}{t}}=
\sum_{k=0}^{\infty}
\mathbf{A}^k (\boldsymbol{\omega}+U_{-k+t}\mathbf{b})\eqsp.
\end{equation*}
%Now let $\theta=(\omega,\chunk{a}{1}{p},\chunk{b}{1}{p})$ and $\thv=(\omega^\star,\chunk{a^\star}{1}{p},\chunk{b^\star}{1}{p})$. Thus by stationarity, if $\tilde\PP^\thv\as$, $\Proj{p}{\F[\theta]{\chunk{U}{-\infty}{0}}}=\Proj{p}{\F[\thv]{\chunk{U}{-\infty}{0}}}$, then for all $t\in\zset$,
%\begin{align}\label{eq:stationary:psi:theta:thv:ident:od:gen}
%\Proj{p}{\F[\theta]{\chunk{U}{-\infty}{t}}}=\Proj{p}{\F[\thv]{\chunk{U}{-\infty}{t}}},\quad\tilde\PP^\thv\as
%\end{align}
%Note also that for all $\theta\in\Theta$, we have for all $t\in\zset$,
%\begin{align*}
%\Proj{p}{\F[\theta]{\chunk{U}{-\infty}{t}}}=\omega+\sum_{k=0}^pa_k\Proj{p}{\F[\theta]{\chunk{U}{-\infty}{-k-1+t}}}+\sum_{k=0}^pb_kU_{-k+t}
%\end{align*}
%It then follows from \eqref{eq:stationary:psi:theta:thv:ident:od:gen} that for all $t\in\zset$, $\tilde\PP^\thv\as$,
%\begin{align*}
%&\omega+\sum_{k=0}^pa_k\Proj{p}{\F[\theta]{\chunk{U}{-\infty}{-k-1+t}}}+\sum_{k=0}^pb_kU_{-k+t}\\
%&=\omega^\star+\sum_{k=0}^pa_k^\star\Proj{p}{\F[\thv]{\chunk{U}{-\infty}{-k-1+t}}}+\sum_{k=0}^pb_k^\star U_{-k+t}.
%\end{align*}
%This implies $\tilde\PP^\thv\as$,
%\begin{align}\label{eq:deduce:ident:theta:thv:od:gen}
%\begin{split}
%(b_1^\star-b_1)U_{t} &=(\omega-\omega^\star)+\sum_{k=0}^p(a_k-a_k^\star)\Proj{p}{\F[\thv]{\chunk{U}{-\infty}{-k-1+t}}} \\
%&+\sum_{k=1}^{p}(b_k-b_k^\star) U_{-k+t}.
%\end{split}
%\end{align}
Moreover, if
\begin{align*}
\F[\theta]{\chunk{U}{-\infty}{0}}=\F[\thv]{\chunk{U}{-\infty}{0}},\quad\tilde\PP^\thv\as,
\end{align*}
then by stationarity, for all $t\in\zset$,
\begin{align*}
\F[\theta]{\chunk{U}{-\infty}{t}}=\F[\thv]{\chunk{U}{-\infty}{t}},\quad\tilde\PP^\thv\as,
\end{align*}
and thus
\begin{equation}\label{eq:ident:obtain:A:k=0:gen:od}
\sum_{k=0}^{\infty}
\mathbf{A}^k (\boldsymbol{\omega}+U_{-k+t}\mathbf{b})=\sum_{k=0}^{\infty}\mathbf{A}_\star^k (\boldsymbol{\omega}_\star+U_{-k+t}\mathbf{b}_\star), \quad\tilde\PP^\thv\as
\end{equation}
This implies
\begin{equation}\label{eq:ident:obtain:A:logpois:gen:od}
(\mathbf{b}-\mathbf{b}_\star)U_{t}=\sum_{k=1}^{\infty}\mathbf{A}_\star^k (\boldsymbol{\omega}_\star+U_{-k+t}\mathbf{b}_\star)-\sum_{k=1}^{\infty}\mathbf{A}^k (\boldsymbol{\omega}+U_{-k+t}\mathbf{b}), \quad\tilde\PP^\thv\as
\end{equation}
Conditionally on $\sigma(\chunk{U}{-\infty}{t-1})$, $\left(\rme^{U_t}-1\right)$ is a Poisson variable with positive intensity. Thus the left-hand side of \eqref{eq:ident:obtain:A:logpois:gen:od} is $\tilde\PP^\thv\as$ constant only if $\mathbf{b}=\mathbf{b}_\star$, implying $b_1^\star=b_1$.
Then, \eqref{eq:ident:obtain:A:k=0:gen:od} reduces to
\begin{equation}\label{eq:ident:obtain:A:k=1:gen:od}
\sum_{k=1}^{\infty}
\mathbf{A}^k (\boldsymbol{\omega}+U_{-k+t}\mathbf{b})=\sum_{k=1}^{\infty}\mathbf{A}_\star^k (\boldsymbol{\omega}_\star+U_{-k+t}\mathbf{b}_\star), \quad\tilde\PP^\thv\as
\end{equation}
Thus by repeating the same argument as to obtain $\mathbf{b}=\mathbf{b}_\star$, \eqref{eq:ident:obtain:A:k=1:gen:od} and so on yield that, for all integer $k\ge1$,
\begin{equation*}%\label{eq:AA:star:b:gen:od}
(\mathbf{A}^k-\mathbf{A}_\star^k)\mathbf{b}_\star=0.
\end{equation*}
And by \eqref{eq:AA:star:b:gen:od}, we have $\mathbf{A}=\mathbf{A}_\star$, yielding that $\left(\chunk{a}{1}{p},\chunk{b}{2}{p}\right)=\left(\chunk{a^\star}{1}{p},\chunk{b^\star}{2}{p}\right)$. It is immediate from \eqref{eq:ident:obtain:A:k=0:gen:od} that $\boldsymbol{\omega}=\boldsymbol{\omega}_\star$, and this is equivalent to having that $\omega=\omega^\star$. The strong consistency therefore follows.
\smallskip \\

{\bf{Proof of \autoref{theo:ergo-convergence:logpois:gen}--\ref{item:thm:logpois:asymp:normal:gen:od}}}. By \autoref{thm:asym:normal:od:gen}, it suffices to show that ~\ref{assum:phi:linear:od}--\ref{item:inverse:fisher:gen:od} hold. From the definition of $\Psi^\theta_u(z)$, we have
$a(\theta,u)=\mathbf{A}$ and $b(\theta,u)=\boldsymbol{\omega}+u\mathbf{b}$, both admitting linear forms in $\theta$. Thus from compactness of $\Theta$ and ~\eqref{eq:def:par:set:logpois:gen} and by noting that for all $\tau>0$,
\begin{equation}\label{eq:moment:finite:expo:logpois:gen:od}
\tilde\PE^\thv\left[\rme^{\tau U_1}\right]<\infty\eqsp,
\end{equation}
it follows that~\ref{assum:phi:linear:od} holds. Assumptions \ref{assum:differentiability:g:od} and \ref{assum:interior:theta:star} clearly hold. Assumption~\ref{assum:strong:consistent} is immediate as the conclusion of  \autoref{theo:ergo-convergence:logpois:gen}-\ref{item:thm:logpois:strong:consistency:gen:od} above. We have
\begin{align*}
\ell_{z,k}^\theta&=\ln g^\theta\left(\Proj{p}{\F{\chunk{U}{1}{k-1}}(z)} ;U_k\right)\\
			&= (\rme^{U_k}-1)\boldsymbol\delta_p^T\F{\chunk{U}{1}{k-1}}(z)-\rme^{ \boldsymbol\delta_p^T\F{\chunk{U}{1}{k-1}}(z)}-\ln(\rme^{U_k}-1)!,\\
			\;\\
\ell_{k}^\theta&=\ln g^\theta\left(\Proj{p}{\F{\chunk{U}{-\infty}{k-1}}(z)} ;U_k\right)\\
			&=(\rme^{U_k}-1)\boldsymbol\delta_p^T\F{\chunk{U}{-\infty}{k-1}}-\rme^{ \boldsymbol\delta_p^T\F{\chunk{U}{-\infty}{k-1}}}-\ln(\rme^{U_k}-1)!.
\end{align*}
Then from \autoref{lem:exist:deriv:psi:od}, for all $i$, $\PPY\as$,
\begin{align*}
\frac{\partial\ell_{z,k}^\theta}{\partial\theta_i}&=\left(\rme^{U_k}-1-\rme^{ \boldsymbol\delta_p^T\F{\chunk{U}{1}{k-1}}(z)}\right) \boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{U}{1}{k-1}}(z) \\
\;\\
\frac{\partial\ell_{k}^\theta}{\partial\theta_i}&=\left(\rme^{U_k}-1-\rme^{ \boldsymbol\delta_p^T\F{\chunk{U}{-\infty}{k-1}}} \right)\boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{U}{-\infty}{k-1}}
\end{align*}
Thus for all $i$, $\PPY\as$,

\begin{align*}
&\left|\frac{\partial\ell_{z,k}^\theta}{\partial\theta_i}-\frac{\partial\ell_{k}^\theta}{\partial\theta_i}\right|
%\le
%\left| \boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{u}{-\infty}{k-1}}- \boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{u}{1}{k-1}}(z)\right|(\rme^{u_k}-1) \\
%&+\left|\rme^{ \boldsymbol\delta_p^T\F{\chunk{u}{-\infty}{k-1}}} \boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{u}{-\infty}{k-1}}-\rme^{ \boldsymbol\delta_p^T\F{\chunk{u}{1}{k-1}}(z)} \boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{u}{1}{k-1}}(z)\right| (\rme^{u_k}-1)\\
\le \left| \boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{U}{-\infty}{k-1}}- \boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{U}{1}{k-1}}(z)\right|(\rme^{U_k}-1) \\
&+\left|\rme^{ \boldsymbol\delta_p^T\F{\chunk{U}{-\infty}{k-1}}}-\rme^{ \boldsymbol\delta_p^T\F{\chunk{U}{1}{k-1}}(z)}\right|\left| \boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{U}{-\infty}{k-1}}\right| \\
&+\rme^{\left| \boldsymbol\delta_p^T\F{\chunk{U}{1}{k-1}}(z)\right|}\left| \boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{U}{-\infty}{k-1}}- \boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{U}{1}{k-1}}(z)\right| \\
\end{align*}
Note that for all $z\in\Zset$ and $k\ge 1$, we have, $\PPY\as$,
\begin{align*}
\F{\chunk{U}{-\infty}{k-1}}-\F{\chunk{U}{1}{k-1}}(z)=\mathbf{A}^{k-1}\left(\F{\chunk{U}{-\infty}{0}}-z\right)\eqsp,
\end{align*}
and for all $i$,
\begin{align*}
\frac{\partial}{\partial\theta_i}\F{\chunk{U}{-\infty}{k-1}}&-\frac{\partial}{\partial\theta_i}\F{\chunk{U}{1}{k-1}}(z) \\
&=\frac{\partial(\mathbf{A}^{k-1})}{\partial\theta_i}\left(\F{\chunk{U}{-\infty}{0}}-z\right)+\mathbf{A}^{k-1}\frac{\partial}{\partial\theta_i}\F{\chunk{U}{-\infty}{0}}\eqsp.
\end{align*}
By similar arguments as in the proof of \autoref{lem:exist:deriv:psi:od} (see~\autoref{sec:proof:lem:deriv:psi:od:gen}) and noting that $\sup_{\theta\in\Theta}\left|\F{\chunk{U}{-\infty}{0}}\right|_\infty$ and $\sup_{\theta\in\Theta}\left|\frac{\partial}{\partial\theta}\F{\chunk{U}{-\infty}{0}}\right|_\infty$ are finite $\tilde\PP^\thv\as$, there exist a constant $\rho_1\in(0,1)$ and a $\tilde\PP^\thv\as$ finite random variable $\tilde M_1=\tilde M_1(z)$ such that for all $k\ge 1$, $\tilde\PP^\thv\as$,
\begin{align*}
\left|\F{\chunk{U}{-\infty}{k-1}}-\F{\chunk{U}{1}{k-1}}(z)\right|_\infty\le \tilde M_1\rho_1^k\eqsp,
\end{align*}
and for all $i$, $\tilde\PP^\thv\as$,
\begin{align*}
\left|\frac{\partial}{\partial\theta_i}\F{\chunk{U}{-\infty}{k-1}}-\frac{\partial}{\partial\theta_i}\F{\chunk{U}{1}{k-1}}(z)\right|_\infty\le \tilde M_1\rho_1^k\eqsp.
\end{align*}
%We also have
%\begin{align}\label{eq:moment:finite:psi:logpois:od:gen}
%\tilde\PE^\thv\left[\sup_{\theta\in\Theta}\left|\F{\chunk{U}{-\infty}{0}}\right|\right]<\infty
%\end{align}
%and for all $i$
From  \autoref{lem:exist:deriv:psi:od}, we also have that, for all $i$,
\begin{align}\label{eq:moment:finite:deriv:psi:logpois:od:gen}
\tilde\PE^\thv\left[\sup_{\theta\in\Theta}\left|\frac{\partial}{\partial\theta_i}\F{\chunk{U}{-\infty}{0}}\right|_\infty\right]<\infty\eqsp.
\end{align}
Recall that for all $t,t'\in\rset$, $\left|\rme^t-\rme^{t'}\right|\le\rme^{|t|\vee |t'|}|t-t'|$.
Thus, we deduce that
\begin{align*}
&\left|\frac{\partial\ell_{z,k}^\theta}{\partial\theta_i}-\frac{\partial\ell_{k}^\theta}{\partial\theta_i}\right| \\
&\le\rme^{\tilde M_1}\rho_1^k\tilde M_1\left[\rme^{U_k}-1+\rme^{\left|\F{\chunk{U}{-\infty}{k-1}}\right|_\infty}\left(1+\left|\frac{\partial}{\partial\theta_i}\F{\chunk{U}{-\infty}{k-1}}\right|_\infty\right)\right]\eqsp.
\end{align*}
On the other hand, $\tilde\PP^\thv\as$, we have
\begin{align*}
\rme^{\left|\F{\chunk{U}{-\infty}{0}}\right|_\infty}\le\rme^{\sup_{\theta\in\Theta}\left|\F{\chunk{U}{-\infty}{0}}\right|_\infty}\le\rme^{\sum_{k=0}^\infty\sup_{\theta\in\Theta}\left|\mathbf{A}^k (\boldsymbol{\omega}+U_{-k}\mathbf{b})\right|_\infty}\eqsp.
\end{align*}
From ~\eqref{eq:norm-control:logpois:gen:od} and \eqref{eq:set:bound:lpois:consist:gen:od}, there exist constants $c_1, c_2>0$ and $\bar\delta\in(0,1)$ such that, $\tilde\PP^\thv\as$,
\begin{align*}
\rme^{\sup_{\theta\in\Theta}\left|\F{\chunk{U}{-\infty}{0}}\right|_\infty}\le c_1\rme^{c_2\sum_{k=0}^\infty\bar\delta^kU_{-k}}\le c_1(1-\bar\delta)\sum_{k=0}^{\infty}\bar\delta^k\rme^{c_2(1-\bar\delta)^{-1}U_{-k}}\eqsp,
\end{align*}
and from \eqref{eq:moment:finite:expo:logpois:gen:od}, it follows that for any positive $t>0$,
\begin{align}\label{eq:finite:moment:exp:phi:lpois:gen:od}
\tilde\PE^\thv\left[\rme^{t\sup_{\theta\in\Theta}\left|\F{\chunk{U}{-\infty}{0}}\right|_\infty}\right]<\infty\eqsp.
\end{align}
Thus, by stationarity and from \eqref{eq:moment:finite:deriv:psi:logpois:od:gen} and \eqref{eq:finite:moment:exp:phi:lpois:gen:od}, \autoref{lem:useful:gen} assures that there exist a constant $\rho_2\in(0,1)$ and a $\tilde\PP^\thv\as$ finite random variable $\tilde M_2=\tilde M_2(z)$ such that for all $i$ and for all $k\ge 1$, $\tilde\PP^\thv\as$,
\begin{align*}
&\left|\frac{\partial\ell_{z,k}^\theta}{\partial\theta_i}-\frac{\partial\ell_{k}^\theta}{\partial\theta_i}\right| \le
\rho_2^k\tilde M_2\eqsp.
\end{align*}
Therefore, Assumption~\ref{item:ctl:limit:od:gen} holds. We now check \ref{item:finte:mement:gen:od}. We have
\begin{align*}
\ell_{1}^\theta&=\ln g^\theta\left(\Proj{p}{\F{\chunk{U}{-\infty}{0}}} ;U_1\right)\\
			&= (\rme^{U_1}-1)\boldsymbol\delta_p^T\F{\chunk{U}{-\infty}{0}}-\rme^{ \boldsymbol\delta_p^T\F{\chunk{U}{-\infty}{0}}}-\ln(\rme^{U_1}-1)!.
\end{align*}
Then for all $i$ and $j$, we have
\begin{align*}
\frac{\partial\ell_{1}^\theta}{\partial\theta_i}	&=\left(\rme^{U_1}-1-\rme^{ \boldsymbol\delta_p^T\F{\chunk{U}{-\infty}{0}}}\right) \boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{U}{-\infty}{0}}
\end{align*}
and from \autoref{lem:exist:deriv:psi:od}, $\PPY\as$,
\begin{align*}
\frac{\partial^2\ell_{1}^\theta}{\partial\theta_i\partial\theta_j}&=\left(\rme^{U_1}-1-\rme^{ \boldsymbol\delta_p^T\F{\chunk{U}{-\infty}{0}}}\right) \boldsymbol\delta_p^T\frac{\partial^2}{\partial\theta_i\partial\theta_j}\F{\chunk{U}{-\infty}{0}} \\
&+\rme^{ \boldsymbol\delta_p^T\F{\chunk{U}{-\infty}{0}}}\left( \boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{U}{-\infty}{0}}\right)\left( \boldsymbol\delta_p^T\frac{\partial}{\partial\theta_j}\F{\chunk{U}{-\infty}{0}}\right)\eqsp.
\end{align*}
Using similar arguments as the proof of \autoref{lem:exist:deriv:psi:od} together with \eqref{eq:moment:finite:expo:logpois:gen:od}, we can show that for all $i$ and $j$, we have
\begin{align*}
\tilde\PE^\thv\left[\sup_{\theta\in\Theta}\left|\frac{\partial}{\partial\theta_i}\F{\chunk{U}{-\infty}{0}}\right|_\infty^2\right]<\infty
\end{align*}
and
\begin{align*}
\tilde\PE^\thv\left[\sup_{\theta\in\Theta}\left|\frac{\partial^2}{\partial\theta_i\partial\theta_j}\F{\chunk{U}{-\infty}{0}}\right|_\infty\right]<\infty.
\end{align*}
By \eqref{eq:finite:moment:exp:phi:lpois:gen:od} and \eqref{eq:moment:finite:expo:logpois:gen:od}, we can directly show that, for all $i$ and $j$,
\begin{align*}
\tilde\PE^\thv\left[\sup_{\theta\in\Theta}\left|\frac{\partial\ell_{1}^\theta}{\partial\theta_i}\right|^2\right]<\infty
\end{align*}
and
\begin{align*}
\tilde\PE^\thv\left[\sup_{\theta\in\Theta}\left|\frac{\partial^2\ell_{1}^\theta}{\partial\theta_i\partial\theta_j}\right|\right]<\infty\eqsp.
\end{align*}
Thus, \ref{item:finte:mement:gen:od}-\ref{item:1:finte:moment:od:gen} and \ref{item:finte:mement:gen:od}-\ref{item:2:finte:moment:od:gen} hold. Note that for all $i, j$, we have, $\PPY\as$,
\begin{align*}
\frac{\partial}{\partial\theta_i}g^\theta\left(\Proj{p}{\F{\chunk{U}{-\infty}{0}}} ;U_1\right)=g^\theta\left(\Proj{p}{\F{\chunk{U}{-\infty}{0}}} ;U_1\right)\frac{\partial\ell_{1}^\theta}{\partial\theta_i}
\end{align*}
and
\begin{align*}
&\frac{\partial^2}{\partial\theta_i\partial\theta_j}g^\theta\left(\Proj{p}{\F{\chunk{Y}{-\infty}{0}}} ;U_1\right) \\
&=g^\theta\left(\Proj{p}{\F{\chunk{U}{-\infty}{0}}} ;U_1\right)\frac{\partial^2\ell_{1}^\theta}{\partial\theta_i\partial\theta_j}+\frac{\partial}{\partial\theta_i}g^\theta\left(\Proj{p}{\F{\chunk{U}{-\infty}{0}}} ;Y_1\right)\frac{\partial\ell_{1}^\theta}{\partial\theta_j}
\end{align*}
Then, \ref{item:finte:mement:gen:od}-\ref{item:3:finte:moment:od:gen} follows by observing that for all $i$, $\PPY\as$,
\begin{align*}
\left|\frac{\partial}{\partial\theta_i}g^\theta\left(\Proj{p}{\F{\chunk{U}{-\infty}{0}}} ;U_1\right)\right|&=g^\theta\left(\Proj{p}{\F{\chunk{U}{-\infty}{0}}} ;U_1\right)\left|\frac{\partial\ell_{1}^\theta}{\partial\theta_i}\right|\le\sup_{\theta\in\Theta}\left|\frac{\partial\ell_{1}^\theta}{\partial\theta_i}\right|
\end{align*}
and
\begin{align*}
\tilde\PE^\thv\left[\sup_{\theta\in\Theta}\left|\frac{\partial\ell_{1}^\theta}{\partial\theta_i}\right|\right]\le\left(\tilde\PE^\thv\left[\sup_{\theta\in\Theta}\left|\frac{\partial\ell_{1}^\theta}{\partial\theta_i}\right|^2\right]\right)^{1/2}<\infty\eqsp.
\end{align*}
Similarly, \ref{item:finte:mement:gen:od}-\ref{item:4:finte:moment:od:gen} holds. To complete the proof of asymptotic normality, it remains to check \ref{item:inverse:fisher:gen:od}. By \autoref{rem:hessian:invet:gen:od}, it is sufficient to show that \eqref{eq:rem:linear:independent:d-logpois:gen:od} holds. We establish this fact by following the argument provided by \cite{ francq2004maximum}. Now let $\boldsymbol{\alpha}=(\alpha_0,\ldots,\alpha_{2p})\in\rset^{2p+1}$ and for convenience, write $\theta=(\theta_0,\ldots,\theta_{2p})=(\omega,\chunk{a}{1:p},\chunk{b}{1}{p})$. We have, $\PPY\as$,
\begin{align*}
\boldsymbol{\alpha}^T\frac{\partial\ell_{1}^\thv}{\partial\theta}=\sum_{i=0}^{2p}\alpha_i\left(\rme^{U_1}-1-\rme^{ \boldsymbol\delta_p^T\F[\thv]{\chunk{U}{-\infty}{0}}}\right) \boldsymbol\delta_p^T\frac{\partial\F[\thv]{\chunk{U}{-\infty}{0}}}{\partial\theta_i}\eqsp.
\end{align*}
Since $\left(\rme^{U_1}-1\right)$ conditionally on $\sigma(\chunk{U}{-\infty}{0})$ is a Poisson variable with positive intensity, then $\tilde\PP^\thv\as$, $\boldsymbol{\alpha}^T\frac{\partial\ell_{1}^\thv}{\partial\theta}=0$  implies
\begin{align*}%\label{eq:lin:indep:1:gen:od}
\sum_{i=0}^{2p}\alpha_i \boldsymbol\delta_p^T\frac{\partial\F[\thv]{\chunk{U}{-\infty}{0}}}{\partial\theta_i}=0,\quad\tilde\PP^\thv\as
\end{align*}
By stationarity, we have for all $t\in\zset$,
\begin{align}\label{eq:lin:indep:stationary:gen:od}
\sum_{i=0}^{2p}\alpha_i \boldsymbol\delta_p^T\frac{\partial\F[\thv]{\chunk{U}{-\infty}{t}}}{\partial\theta_i}=0,\quad\tilde\PP^\thv\as
\end{align}
In view of \eqref{eq:def:logpois:gen:trans} and by noting that $\PPY\as$, $X_{t}^\theta=\boldsymbol\delta_p^T\F[\theta]{\chunk{U}{-\infty}{t}}$ for all $\theta\in\Theta$ and $t\in\zset$, we have from \eqref{eq:lin:indep:stationary:gen:od} that for all $t\in\zset$,

\begin{align*}
0=\boldsymbol\alpha^T \frac{\partial X_{t}^\thv}{\partial\theta}=\boldsymbol\alpha^T \left(\begin{array}{c}1 \\X_{t-1}^\thv \\\vdots \\X_{t-p}^\thv \\U_{t-1} \\\vdots \\U_{t-p}\end{array}\right)+\sum_{i=1}^{p}a_i\boldsymbol\alpha^T\frac{\partial X_{t-i}^\thv}{\partial\theta}= \boldsymbol\alpha^T \left(\begin{array}{c}1 \\X_{t-1}^\thv \\\vdots \\X_{t-p}^\thv \\U_{t-1} \\\vdots \\U_{t-p}\end{array}\right), \tilde\PP^\thv\as
\end{align*}
%
%Note that we have
%\begin{align}\label{eq:lin:indep:2:gen:od}
%\frac{\partial\F[\thv]{\chunk{U}{-\infty}{0}}}{\partial\theta_{2p}}=\sum_{k=1}^{\infty}U_{-k}\mathbf{A}_\star^k\boldsymbol{\delta}_p+U_0\boldsymbol{\delta}_p,\quad\tilde\PP^\thv\as
%\end{align}
Since $\left(\rme^{U_{t-1}}-1\right)$ conditionally on $\sigma(\chunk{U}{-\infty}{t-2})$ is a Poisson variable with positive intensity, we then have that $\alpha_{p+1}=0$. By the same reason, it can be shown that $\chunk{\alpha}{p+2}{p+2+i}=0$ if $\chunk{\alpha}{2}{2+i}=0$, for $i\in\{0,\ldots,p-2\}$. Thus, if $\boldsymbol\alpha\neq0$ entails an \acrshort{odm}$(p-1,p-1)$ representation, which is impossible since the model is identifiable. Hence, $\boldsymbol\alpha=0$ and this completes the proof.
%from \eqref{eq:lin:indep:1:gen:od} and \eqref{eq:lin:indep:2:gen:od} that $\alpha_{2p}=0$.
%Following similar argument, we get, for all $k\ge 1$,
%\begin{align*}%\label{eq:invert:A:Astar:zero:gen:od}
%\sum_{i=1}^{2p-1}\alpha_i\boldsymbol{\delta}_p^T\left(\left.\frac{\partial\mathbf{A}^k}{\partial\theta_i}\right|_{\mathbf{A}=\mathbf{A}_\star}\right)\mathbf{b}_\star=0\eqsp.
%\end{align*}
%By \eqref{eq:invert:A:Astar:zero:gen:od}, it yields $\chunk{\alpha}{1}{2p-1}=0$. Then from \eqref{eq:lin:indep:1:gen:od}, we finally have
%\begin{align*}
%\alpha_0\boldsymbol\delta_p^T\frac{\partial}{\partial\theta_0}\F[\thv]{\chunk{U}{-\infty}{0}}=\alpha_0\boldsymbol\delta_p^T\left(\mathbf{I}_{2p-1}-\mathbf{A}_\star\right)^{-1}\boldsymbol\delta_p=0\eqsp,
%\end{align*}
%which implies $\alpha_0=0$ since $\boldsymbol\delta_p^T\left(\mathbf{I}_{2p-1}-\mathbf{A}_\star\right)^{-1}\boldsymbol\delta_p=(1-\sum_{i=1}^pa_i^\star)^{-1}\neq0$.
\end{proof}

%%%%%%%%%%%%%%%%%%
%In this section, we will use the vector and the matrix norms defined respectively on $\rset^{2p-1}$ and the space of $(2p-1)\times (2p-1)$ matrices as follows:
%$$
%|z|_\infty=\bigvee_{i=1}^{2p-1}|\boldsymbol{\delta}_i^Tz|=\bigvee_{i=1}^{2p-1}|z_i|\eqsp,
%$$
%$$
%\|\mathbf{M}\|_\infty= \bigvee_{i=1}^{2p-1} \sum_{j=1}^{2p-1} |\mathbf{M}_{i,j}|=\sup_{|z|_\infty\leq1}\left|\mathbf{M}z\right|_\infty.
%$$
%For notational convenience, we will also denote $\mathbf{A}_1=\mathbf{A}$, $\mathbf{A}_2=\mathbf{b}\boldsymbol{\delta}_p^T$ and $\mathbf{A}_3=\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T$.
%
% \begin{proposition}\label{theo:ergo-convergence:logpois:gen}
%  Suppose that $\theta\in\Theta$ satisfies
%  \begin{equation}\label{eq:def:par:set:logpois:gen}
% \bigvee_{(k_1,\ldots,k_p)\in\{1,2,3\}^p}\left\|\prod_{j=1}^p\mathbf{A}_{k_j}\right\|_\infty<1\eqsp.
%  \end{equation}
% Then we have the following.
% \begin{enumerate}[label=(\roman*)]
% \item\label{item:thm:logpois:ergodic:gen:od}
%\autoref{thm:ergodicity:gen} holds with the function $V: \Zset\to\rsetp$ defined by,  for all $z\in\Zset$,
% \begin{equation}\label{eq:def:V:logpois:gen}
% V(z)=\rme^{\tau |z|_\infty}\eqsp,\quad \tau>0\eqsp.
% \end{equation}
% \item\label{item:thm:logpois:strong:consistency:gen:od}
%\autoref{thm:convergence-main:gen:od} holds with any $z_1\in\Zset$. If the true parameter $\thv=(\omega^\star,\chunk{a^\star}{1}{p},\chunk{b^\star}{1}{p})$ moreover satisfies: for all $k\in\zsetpnz$,
%\begin{equation}\label{eq:AA:star:b:gen:od}
%(\mathbf{A}^k-\mathbf{A}_\star^k)\mathbf{b}_\star=0\quad \implies \quad\mathbf{A}=\mathbf{A}_\star\eqsp,
%\end{equation}
%then \autoref{thm:strong:consistency:gen:od} also holds.
%\item\label{item:thm:logpois:asymp:normal:gen:od}
%If the true parameter $\thv$ lies in the interior of $\Theta$ and satisfies \eqref{eq:AA:star:b:gen:od} and, for all $k\in\zsetpnz$ and $\chunk{\alpha}{1}{2p-1}\in\rset^{2p-1}$, writing $\chunk{\theta}{1}{2p-1}=(\chunk{a}{1}{p},\chunk{b}{2}{p})$,
%\begin{align}\label{eq:invert:A:Astar:zero:gen:od}
%\sum_{i=1}^{2p-1}\alpha_i\boldsymbol{\delta}_p^T\left(\left.\frac{\partial\mathbf{A}^k}{\partial\theta_i}\right|_{\mathbf{A}=\mathbf{A}_\star}\right)\mathbf{b}_\star=0 \quad \implies \quad \chunk{\alpha}{1}{2p-1}=0\eqsp,
%\end{align}
%%and
%%\begin{align}\label{eq:invert:omega:zero:gen:od}
%%\boldsymbol{\delta}_p^T(\mathbf{I}_{2p-1}-\mathbf{A}_\star)^{-1}\boldsymbol{\delta}_p\neq0\eqsp,
%%\end{align}
%then \autoref{thm:asym:normal:od:gen} holds.
% \end{enumerate}
%\end{proposition}
%
%\begin{remark}
%To check \eqref{eq:AA:star:b:gen:od}, it suffices to check that there exists $k\in\zsetp$ such that the determinant
%\begin{align*}
%\det\left(\mathbf{A}_\star^k\mathbf{b}_\star,\mathbf{A}_\star^{k+1}\mathbf{b}_\star,\ldots,\mathbf{A}_\star^{k+2p-2}\mathbf{b}_\star\right)\neq0\eqsp.
%\end{align*}
%%is not identically zero, which is in turn equivalent to checking that the determinant
%%\begin{align}\label{eq:det:conjecture:non:zero:gen:od}
%%\left|\begin{array}{ccccc}
%%\boldsymbol\delta_p^T\mathbf{b}_\star & \boldsymbol\delta_p^T\mathbf{A}_\star\mathbf{b}_\star & \cdots & \boldsymbol\delta_p^T\mathbf{A}_\star^{p-2}\mathbf{b}_\star & \boldsymbol\delta_p^T\mathbf{A}_\star^{p-1}\mathbf{b}_\star\\
%%\boldsymbol\delta_p^T\mathbf{A}_\star\mathbf{b}_\star & \boldsymbol\delta_p^T\mathbf{A}_\star^2\mathbf{b}_\star & \cdots & \boldsymbol\delta_p^T\mathbf{A}_\star^{p-1}\mathbf{b}_\star & \boldsymbol\delta_p^T\mathbf{A}_\star^{p}\mathbf{b}_\star\\
%%\vdots & \vdots & \ddots & \vdots & \vdots\\
%%\boldsymbol\delta_p^T\mathbf{A}_\star^{p-2}\mathbf{b}_\star &\boldsymbol\delta_p^T\mathbf{A}_\star^{p-1}\mathbf{b}_\star & \ldots & \boldsymbol\delta_p^T\mathbf{A}_\star^{2p-4}\mathbf{b}_\star & \boldsymbol\delta_p^T\mathbf{A}_\star^{2p-3}\mathbf{b}_\star\\
%%\boldsymbol\delta_p^T\mathbf{A}_\star^{p-1}\mathbf{b}_\star & \boldsymbol\delta_p^T\mathbf{A}_\star^{p}\mathbf{b}_\star & \ldots & \boldsymbol\delta_p^T\mathbf{A}_\star^{2p-3}\mathbf{b}_\star & \boldsymbol\delta_p^T\mathbf{A}_\star^{2p-2}\mathbf{b}_\star
%%\end{array}\right|
%%\end{align}
%%does not identically vanish.
%\end{remark}
%
%\begin{remark}
%When $p=1$, \eqref{eq:AA:star:b:gen:od} and \eqref{eq:invert:A:Astar:zero:gen:od} hold if and only if $b^\star\neq 0$.
%\end{remark}
%\smallskip
%


%\begin{proof} {\bf{Proof of \autoref{theo:ergo-convergence:logpois:gen}--\ref{item:thm:logpois:ergodic:gen:od}}}. From \autoref{thm:ergodicity:gen}, we need to show that ~\ref{ass-ergo-Zset:gen}, \ref{assum:weakFeller-V:gen} with $V$ given in~\eqref{eq:def:V:logpois:gen},  \ref{assum:reachable:gen}, \ref{assum:bound:rho:gen}
%  and~\ref{assum:alpha-phi:gen} hold. Assumption~~\ref{ass-ergo-Zset:gen} holds with the metric $\Zmet$ defined by, for all $(z,z')\in\Zset^2$,
%  \begin{equation}\label{eq:def:zmet:gen:od}
%  \Zmet(z,z')=\bigvee_{i=1}^{2p-1}|z_i-z_i'|=|z-z'|_\infty\eqsp.
%  \end{equation}
% The fact that $R$ is weak Feller follows from the continuity of the density $g$ with respect to the $x$ component in \eqref{eq:density:g:logpois:gen:od} and the continuity of $\Proj{p}{\cdot}$. We will show that, with $V$ as defined in \eqref{eq:def:V:logpois:gen}, then we have $R^pV\le\lambda V+\beta$ for some $(\lambda,\beta)\in(0,1)\times\rsetp$.  To this end, it worths proving the following lemma.
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{lemma}\label{lem:ineq:V':phi'}
% Let $\mathbf{a}\in\rset^{2p-1}$, $\tau>0$ and $V':\Zset\to\rsetp$ be defined by for all $z\in\Zset$, $V'(z)=\rme^{\tau|\mathbf{a}^Tz|}$. Then for any function $\phi': \Zset^2\to\Proj{p}{\Zset}$ such that $\phi'(z,z')\le \Proj{p}{z}$ for all $(z,z')\in\Zset^2$, we have
% \begin{equation*}
%\int V'(\boldsymbol{\omega}+\mathbf{A}z+u\mathbf{b})G(\phi'(z,z');\rmd u)\le\gamma_0\rme^{\tau\left(\left|\mathbf{a}^T\mathbf{A}z\right|\vee\left|\mathbf{a}^T\mathbf{b}\boldsymbol{\delta}_p^Tz\right|\vee\left|\mathbf{a}^T(\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T)z\right|\right)}\eqsp,
% \end{equation*}
% for some $\gamma_0>0$.
%  \end{lemma}
%
% \begin{proof}
% Note that if real numbers $u$ and $v$ have different signs or if $v=0$, then $|u+v|\le |u| \vee |v|$. Otherwise, $|u+v|=(u+v)\1\{v>0\}\vee(-u-v)\1\{v<0\}$. This implies that
% \begin{equation*}
% \rme^{|u+v|} \le \rme^{|u|}+\rme^{|v|}+\rme^{u+v}\1\{v>0\}+\rme^{-u-v} \1\{v<0\} \eqsp.
% \end{equation*}
% Using this inequality, we have for all $z\in\Zset$,
%% For $V'(z)=\rme^{\tau\left|\mathbf{a}^Tz\right|}$ with $|\mathbf{a}^T\mathbf{b}|\le1$, using the above inequality, we have for all $z\in\Zset$,
%\begin{align}
%\nonumber
%&\int V'(\boldsymbol{\omega}+\mathbf{A}z+u\mathbf{b})\;G(\phi'(z,z');\rmd u)=\int\rme^{\tau\left|\mathbf{a}^T(\boldsymbol{\omega}+\mathbf{A}z+u\mathbf{b})\right|}\;G(\phi'(z,z');\rmd u)\\ \nonumber
%&\le
%\rme^{\tau\left|\mathbf{a}^T\boldsymbol{\omega}\right|}\int\left( \rme^{\tau\left|\mathbf{a}^T\mathbf{A}z\right|} + \rme^{\tau\left|\mathbf{a}^T\mathbf{b}u\right|} +  \rme^{\tau\left(\mathbf{a}^T\mathbf{A}z + \mathbf{a}^T\mathbf{b}u\right)}\1\{\mathbf{a}^T\mathbf{b}>0\} \right. \\ \label{eq:RV:temp:gen}
%&
%\left. +\rme^{\tau\left(-\mathbf{a}^T\mathbf{A}z - \mathbf{a}^T\mathbf{b}u\right)}\1\{\mathbf{a}^T\mathbf{b}<0\}\right)G(\phi'(z,z');\rmd u)\eqsp.
%\end{align}
%Note further that, by~\cite{znidaric2005asymptotic}, if $Y\sim\mathcal{P}(\lambda)$ with $\lambda>0$, then for any $\vartheta>0$, there exist constants $c_1=c_1(\vartheta), c_2=c_2(\vartheta)>0$ such that
%$$
%\PE[(1+Y)^\vartheta]\le c_1+c_2\lambda^\vartheta.
%$$
%% This implies, for all $\vartheta>0$ and $z\in\Zset$,
%% $$
%% \int \rme^{\vartheta u}G\left(\boldsymbol{\delta}_p^Tz;\rmd u\right)\le c_1+c_2\rme^{\vartheta\boldsymbol{\delta}_p^Tz}\eqsp,
%% $$
%% for some positive constants $c_1$ and $c_2$.
%Then, we get that for any $\vartheta>0$, there exist some constants $\beta_1=\beta_1(\vartheta),\beta_2=\beta_2(\vartheta)>0$ such that
%\begin{equation}\label{eq:help:trans:exp}
%\int \rme^{\vartheta u}\;G(\phi'(z,z');\rmd u)\le \beta_1\rme^{\vartheta\phi'(z,z')}+\beta_2\le \beta_1\rme^{\vartheta\Proj{p}{z}}+\beta_2\eqsp,
%\end{equation}
%where we have used that $\phi'(z,z')\le\Proj{p}{z}$
%%\begin{equation}\label{eq:help:trans:exp}
%%\int \rme^{\vartheta u}\;G(\phi'(z,z');\rmd u)\le (1+\rme^{\phi'(z,z')})^{\vartheta}\le 1+\rme^{\vartheta\phi'(z,z')}\le 1+\rme^{\vartheta\Proj{p}{z}}\eqsp.
%%\end{equation}
%%Furthermore, since $|\mathbf{a}^T\mathbf{b}|\le1$, we have $\mathbf{a}^T\mathbf{b}\1\{\mathbf{a}^T\mathbf{b}>0\}\in[0,1]$ and $-\mathbf{a}^T\mathbf{b}\1\{\mathbf{a}^T\mathbf{b}<0\}\in[0,1]$.
%Then from \eqref{eq:RV:temp:gen} and \eqref{eq:help:trans:exp}, it follows that there exist constants $c_{ij}>0$, $(i, j)\in\{1,2,3\}\times\{1,2\}$, such that
%
%\begin{align*}
%\int V'(\boldsymbol{\omega}&+\mathbf{A}z+u\mathbf{b})\;G(\phi'(z,z');\rmd u)\\
%&\le
%\rme^{\tau\left|\mathbf{a}^T\boldsymbol{\omega}\right|}\left( \rme^{\tau\left|\mathbf{a}^T\mathbf{A}z\right|} +c_{11}\rme^{\tau\left|\mathbf{a}^T\mathbf{b}\Proj{p}{z}\right|}+c_{12} \right. \\
%&
%\left. +\rme^{\tau\mathbf{a}^T\mathbf{A}z}\left(c_{21}\rme^{\tau\mathbf{a}^T\mathbf{b}\Proj{p}{z}}+c_{22}\right)\1\{\mathbf{a}^T\mathbf{b}>0\} \right. \\
%&
%\left. +\rme^{-\tau\mathbf{a}^T\mathbf{A}z}\left(c_{31}\rme^{-\tau\mathbf{a}^T\mathbf{b}\Proj{p}{z}}+c_{32}\right)\1\{\mathbf{a}^T\mathbf{b}<0\}\right) \\
%&\le
%\rme^{\tau\left|\mathbf{a}^T\boldsymbol{\omega}\right|}\left((1+c_{22}+c_{32})\rme^{\tau\left|\mathbf{a}^T\mathbf{A}z\right|} +c_{11}\rme^{\tau\left|\mathbf{a}^T\mathbf{b}\boldsymbol{\delta}_p^Tz\right|} \right. \\
%&
%\left.+ (c_{21}+c_{31}) \rme^{\tau\left|\mathbf{a}^T(\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T)z\right|} +c_{12}\right) \\
%&\le
%\gamma_0\rme^{\tau\left(\left|\mathbf{a}^T\mathbf{A}z\right|\vee\left|\mathbf{a}^T\mathbf{b}\boldsymbol{\delta}_p^Tz\right|\vee\left|\mathbf{a}^T(\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T)z\right|\right)}\eqsp,
%\end{align*}
%where $\gamma_0=4\left(1+\sum_{i,j}c_{ij}\right)\rme^{\tau\left|\mathbf{a}^T\boldsymbol{\omega}\right|}$.
% \end{proof}
%
% Now, using back and forth the inequalities
%\begin{equation}\label{eq:ineq:exp:norm:gen}
%\rme^{\tau|z|_\infty}\le\sum_{i=1}^{2p-1}\rme^{\tau\left|\boldsymbol{\delta}_i^Tz\right|}\le(2p-1)\rme^{\tau|z|_\infty}\eqsp,
%\end{equation}
%we get, for all $z\in\Zset$,
%\begin{align*}
%RV(z)\le
%\gamma_1\mathbf{e}^{\tau\bigvee_{i=1}^{2p-1}\left(\left|\boldsymbol{\delta}_i^T\mathbf{A}_1z\right|\vee\left|\boldsymbol{\delta}_i^T\mathbf{A}_2z\right|\vee\left|\boldsymbol{\delta}_i^T\mathbf{A}_3z\right|\right)}\eqsp,
%\end{align*}
%for some $\gamma_1>0$.
%By iteration, $R^{j+1}V=R(R^jV)$, and applying again \autoref{lem:ineq:V':phi'} with $\phi'(z,z')=\Proj{p}{z}$, for all $(z,z')\in\Zset^2$, and \eqref{eq:ineq:exp:norm:gen} at each step, we deduce that, for some $\gamma_p>0$, for all $z\in\Zset$,
%\begin{align}\label{eq:ineq:V:Cond}
%\begin{split}
%R^pV(z)\le
%\gamma_p\mathbf{e}^{\tau\bigvee_{i=1}^{2p-1}\left(\bigvee_{k_1,\ldots, k_p\in\{1,2,3\}^p}\left|\boldsymbol{\delta}_i^T\prod_{j=1}^p\mathbf{A}_{k_j}z\right|\right)}
%\le
%\gamma_p\mathbf{e}^{\eta\tau|z|_\infty}
%\eqsp,
%\end{split}
%\end{align}
%where $\eta=\bigvee_{{k_1,\ldots,k_p}\in\{1,2,3\}^p}\left\|\prod_{j=1}^p\mathbf{A}_{k_j}\right\|_\infty<1$.
%%By~\cite{znidaric2005asymptotic}, if $Y\sim\mathcal{P}(\lambda)$ with $\lambda>0$, then for any $\vartheta\in\rset$, there exist constants $c_1=c_1(\vartheta), c_2=c_2(\vartheta)>0$ such that
%%$$
%%\PE[(1+Y)^\vartheta]\le c_1+c_2\lambda^\vartheta.
%%$$
%% This implies, for all $\vartheta\in\rset$ and $z\in\Zset$,
%% $$
%% \int \rme^{\vartheta u}G\left(\boldsymbol{\delta}_p^Tz;\rmd u\right)\le c_1+c_2\rme^{\vartheta\boldsymbol{\delta}_p^Tz}\eqsp,
%% $$
%% for some positive constants $c_1$ and $c_2$. Note further that for all $z\in\Zset$ and $\tau>0$, we have
%% $$V(z)=\rme^{\tau|z|}\le\sum_{(\sigma_{j_1},\ldots,\sigma_{j_{2p-1}})\in\{-1,1\}^{2p-1}}\rme^{\sum_{k=1}^{2p-1}\sigma_{j_k}\tau z_k}\le 2^{2p-1}\rme^{\tau|z|}.$$
%% Without loss of generality, let us assume for the moment that $\tau=1$. Now for all $z\in\Zset$, we have
%% \begin{align*}
%% RV(z)&\le\sum_{(\sigma_{j_1},\ldots,\sigma_{j_{2p-1}})\in\{-1,1\}^{2p-1}}\int \rme^{\sum_{k=1}^{2p-1}\sigma_{j_k}\boldsymbol{\delta}_k^T(\omega+\mathbf{A}z+u\mathbf{b})}G\left(\boldsymbol{\delta}_p^Tz;\rmd u\right)\\
%% &\le\sum_{(\sigma_{j_1},\ldots,\sigma_{j_{2p-1}})\in\{-1,1\}^{2p-1}}\rme^{\sum_{k=1}^{2p-1}\sigma_{j_k}\boldsymbol{\delta}_k^T(\omega+\mathbf{A}z)}\int \rme^{\sum_{k=1}^{2p-1}\sigma_{j_k}\boldsymbol{\delta}_k^T\mathbf{b}u}G\left(\boldsymbol{\delta}_p^Tz;\rmd u\right) \\
%% &\le\sum_{(\sigma_{j_1},\ldots,\sigma_{j_{2p-1}})\in\{-1,1\}^{2p-1}}\rme^{\sum_{k=1}^{2p-1}\sigma_{j_k}\boldsymbol{\delta}_k^T(\omega+\mathbf{A}z)}\left(c_1+c_2 \rme^{\sum_{k=1}^{2p-1}\sigma_{j_k}\boldsymbol{\delta}_k^T\mathbf{b}\boldsymbol{\delta}_p^Tz}\right)\eqsp,
%% \end{align*}
%% for some positive constants $c_1$ and $c_2$. The above inequality can be further simplified to, with $\mathbf{A}_1=\mathbf{A}$ and $\mathbf{A}_2=\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T$,
%%  \begin{align*}
%% RV(z) &\le\sum_{(\sigma_{j_1},\ldots,\sigma_{j_{2p-1}})\in\{-1,1\}^{2p-1}}\left(\beta_1\rme^{\sum_{k=1}^{2p-1}\sigma_{j_k}\boldsymbol{\delta}_k^T\mathbf{A}z}+\beta_2 \rme^{\sum_{k=1}^{2p-1}\sigma_{j_k}\boldsymbol{\delta}_k^T(\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T)z}\right)\eqsp\\
%%& \le\sum_{(\sigma_{j_1},\ldots,\sigma_{j_{2p-1}})\in\{-1,1\}^{2p-1}}\left(\beta_1\rme^{\sum_{k=1}^{2p-1}\sigma_{j_k}\boldsymbol{\delta}_k^T\mathbf{A}_1z}+\beta_2 \rme^{\sum_{k=1}^{2p-1}\sigma_{j_k}\boldsymbol{\delta}_k^T\mathbf{A}_2z}\right)
%% \end{align*}
%%for some positive constants $\beta_1$ and $\beta_2$.
%% By iteration, we obtain that
%%
%%  \begin{align*}
%% R^pV(z) & \le\sum_{(\sigma_{j_1},\ldots,\sigma_{j_{2p-1}})\in\{-1,1\}^{2p-1}}\sum_{(i_1,\ldots,i_p)\in\{1,2\}^p}\beta_{i_1,\ldots,i_p}\rme^{\sum_{k=1}^{2p-1}\sigma_{j_k}\boldsymbol{\delta}_k^T\left(\prod_{\ell=1}^p\mathbf{A}_{i_\ell}\right)z} \\
%% & \le\sum_{(i_1,\ldots,i_p)\in\{1,2\}^p}\beta_{i_1,\ldots,i_p}\sum_{(\sigma_{j_1},\ldots,\sigma_{j_{2p-1}})\in\{-1,1\}^{2p-1}}\rme^{\sum_{k=1}^{2p-1}\sigma_{j_k}\boldsymbol{\delta}_k^T\left(\prod_{\ell=1}^p\mathbf{A}_{i_\ell}\right)z}  \\
%%  & \le\sum_{(i_1,\ldots,i_p)\in\{1,2\}^p}\beta_{i_1,\ldots,i_p}\sum_{(\sigma_{j_1},\ldots,\sigma_{j_{2p-1}})\in\{-1,1\}^{2p-1}}\rme^{\left\|\prod_{\ell=1}^p\mathbf{A}_{i_\ell}\right\||z|} \\
%%   & \le2^{2p-1}\sum_{(i_1,\ldots,i_p)\in\{1,2\}^p}\beta_{i_1,\ldots,i_p}\rme^{\left\|\prod_{\ell=1}^p\mathbf{A}_{i_\ell}\right\||z|} \\
%%   & \le\left(2^{2p-1}\sum_{(i_1,\ldots,i_p)\in\{1,2\}^p}\beta_{i_1,\ldots,i_p}\right)\rme^{\bigvee_{(i_1,\ldots,i_p)\in\{1,2\}^p}\left\|\prod_{\ell=1}^p\mathbf{A}_{i_\ell}\right\||z|},
%% \end{align*}
%% where for all $(i_1,\ldots,i_p)\in\{1,2\}^p$, $\beta_{i_1,\ldots,i_p}$ are positive constants.
%%Thus, for all $z\in\Zset$,
%%$$
%%R^pV(z)\le \gamma_p\rme^{\eta_p|z|},
%%$$
%%where
%%$$\gamma_p\eqdef2^{2p-1}\sum_{(i_1,\ldots,i_p)\in\{1,2\}^p}\beta_{i_1,\ldots,i_p}$$
%%and
%%$$\eta_p\eqdef\bigvee_{(i_1,\ldots,i_p)\in\{1,2\}^p}\left\|\prod_{\ell=1}^p\mathbf{A}_{i_\ell}\right\|<1.$$
%Since
%\begin{equation*}
%\lim_{|z|_\infty\to\infty}\frac{\mathbf{e}^{\eta|z|_\infty}}{V(z)}=0\eqsp
%\end{equation*}
%and for all $M>0$,
%\begin{equation*}
%\sup_{|z|_\infty\le M}R^pV(z)<\infty\eqsp,
%\end{equation*}
%then there exists $(\lambda,\beta)\in(0,1)\times\rsetp$ such that $R^pV\le\lambda V+\beta$, and hence~\ref{assum:weakFeller-V:gen}
%holds. We now prove \ref{assum:reachable:gen}.
%Note that
%from~\eqref{eq:def:par:set:logpois:gen}, there exists $\delta\in(0,1)$ and $c>0$ such
%that, for any $k\geq1$,
%\begin{equation}
%  \label{eq:norm-control:logpois:gen:od}
%\left\|\mathbf{A}^k\right\|_\infty\leq c\;
%\delta^k \;.
%\end{equation}
%Hence $(\mathbf{I}_{2p-1}-\mathbf{A})^{-1}=\mathbf{I}_{2p-1}+\sum_{k\ge1}\mathbf{A}^k$ is well defined. Now set
%$z_\infty=(\mathbf{I}_{2p-1}-\mathbf{A})^{-1}\boldsymbol{\omega}$. Let $\mathcal{N}_\infty$ be an open neighborhood of $z_\infty$ and let $z\in\Zset$. Define recursively the sequence $z_0 = z$ and for all $k\ge1$, $z_k=\boldsymbol{\omega}+\mathbf{A}z_{k-1}$. From~\eqref{eq:norm-control:logpois:gen:od}, this so-defined sequence admits a unique limiting point $\lim_{n\to\infty}z_n=z_\infty$. Thus, there exists some $n\in\zsetpnz$ such that for all $k\ge n$, $z_k\in\mathcal{N}_\infty$. For such $n$, we have
%\begin{align*}
%R^n(z,\mathcal{N}_\infty)=\bar\PP_{z}(Z_n\in\mathcal{N}_\infty)&\ge\bar\PP_{z}(Z_k=z_k \,\, \text{for all} \,\, k=1,\ldots,n)\\
%&=\bar\PP_{z}(U_0=\cdots=U_{n-1}=0)>0\eqsp.
%\end{align*}
%
%Next we prove~\ref{assum:bound:rho:gen}. We have for all $n\geq1$, $u_{0:n-1}\in\Uset^{n}$ and $z\in\Zset$,
%\begin{equation}\label{eq:psi:logpois:iterate:gen:od}
%\F[]{\chunk{u}{0}{n-1}}(z)=
%\mathbf{A}^{n}z+\sum_{j=0}^{n-1}
%\mathbf{A}^j (\boldsymbol{\omega}+u_{n-1-j}\mathbf{b})\eqsp.
%\end{equation}
%Then for all $(z,z')\in\Zset^2$, $\chunk{u}{0}{p-1}\in\Uset^p$,
%\begin{align*}
%\Zmet(\Psi_{u_0}(z),\Psi_{u_0}(z'))&=|\mathbf{A}(z-z')|_\infty \\
%&\le \|\mathbf{A}\|_\infty|z-z'|_\infty\\
%&\le D\Zmet(z,z')\eqsp,
%\end{align*}
%and
%\begin{align*}
%\Zmet(\F[]{\chunk{u}{0}{p-1}}(z),\F[]{\chunk{u}{0}{p-1}}(z'))&=|\mathbf{A}^p(z-z')|_\infty \\
%&\le \|\mathbf{A}^p\|_\infty|z-z'|_\infty\\
%&\le \rho\Zmet(z,z')\eqsp,
%\end{align*}
%where $D=\|\mathbf{A}\|_\infty>0$ and $\rho=\|\mathbf{A}^p\|_\infty<1$ from~\eqref{eq:def:par:set:logpois:gen}, showing ~\ref{assum:bound:rho:gen}.
% To show~\ref{assum:alpha-phi:gen}, we rely
%on~\autoref{lem:det:alpha:gen:od}. Let us set $\Cset=\rset=\Xset$ and
%$\Sset=\{1\}$, then the probability measure $\mu$ boils down the Dirac mass on $\Sset$. For all $(x,u)\in\Xset\times\Uset$, let $j(x)=\rme^{-\rme^x}$ and
%$h(x;u)=\frac{\rme^{x(\rme^u-1)}}{(\rme^u-1)!}$. Indeed,
%$h$ satisfies ~\ref{item:F1:gen}. Hence, from lemma~\autoref{lem:det:alpha:gen:od} we
%$\alpha$ and $\phi$ are obtained by setting: for all $z,  z'\in\Zset$,
%$$\alpha(z,z')=\frac{\rme^{-\rme^{\Pi_{p}(z)\vee\Pi_{p}(z')}}}{\rme^{-\rme^{\Pi_{p}(z)\wedge\Pi_{p}(z')}}}
%=\rme^{-\left|\rme^{\Pi_p(z)}-\rme^{\Pi_p(z')}\right|}
%$$
%and $\phi(z,z')=\Proj{p}{z}\wedge \Proj{p}{z'}$. These so-defined functions $\alpha$ and $\phi$ indeed satisfy ~\ref{assum:alpha-phi:gen}\ref{item:alpha-phi:3:gen}.  For $(z, z')\in\Zset^2$, we have
%\begin{align*}
%1-\alpha(z,z') &=1-\rme^{-\left|\rme^{\Pi_p(z)}-\rme^{\Pi_p(z')}\right|}\le \left|\rme^{\Pi_p(z)}-\rme^{\Pi_p(z')}\right|\\
%&\leq  \rme^{|\Proj{p}{z}\vee\Proj{p}{z'}|}\left|\Proj{p}{z}-\Proj{p}{z'}\right|\\
%& \leq W(z,z')\;\Zmet(z,z') \;,
%\end{align*}
%where $W$ is defined by, for all $z,z'\in\Zset$,
%$$W(z,z')\eqdef\rme^{|z|_\infty\vee|z'|_\infty}\eqsp.$$
%Then
%~\ref{assum:alpha-phi:gen}\ref{assum:definition-gamma-x:gen} and
%~\ref{assum:alpha-phi:gen}\ref{assum:hyp:1-alpha:W:gen} hold. To complete the proof of \autoref{theo:ergo-convergence:logpois:gen}--\ref{item:thm:logpois:ergodic:gen:od}, it remains to show ~\ref{assum:alpha-phi:gen}\ref{assum:driftCond:W:new:gen:od}. Now let $V_1(z)=\rme^{|z|_\infty}$. Note that \ref{assum:weakFeller-V:gen} holds with $V(z)=\rme^{\tau|z|_\infty}$ for arbitrary $\tau>0$, thus it also holds with $V=V_1$. Observe moreover that, for all $z,z'\in\Zset$,
%$$
%W(z,z')=V_1(z)\vee V_1(z')\le V_1(z)+V_1(z')\eqsp.
%$$
%Following the same lines as to obtain \eqref{eq:ineq:V:Cond},  applying \autoref{lem:ineq:V':phi'} with $\tau=1$ and $\phi'=\phi$ and using \eqref{eq:ineq:exp:norm:gen} back and forth, we deduce that, for each $n\in\{1,\dots, p\}$, there exists $\gamma_n>0$ such that for all $z\in\Zset$,
%\begin{equation*}
%J_n((z,z');W)\le \gamma_n W(z,z')^{\vartheta_n}\le DW(z,z')^{\vartheta_n}\eqsp,
%\end{equation*}
%where $D= \bigvee_{i=1}^p\gamma_n>0$ and $\vartheta_n=\bigvee_{{k_1,\ldots,k_n}\in\{1,2,3\}^n}\left\|\prod_{j=1}^n\mathbf{A}_{k_j}\right\|_\infty$.
%Thus \ref{assum:alpha-phi:gen}\ref{assum:driftCond:W:new:gen:od} follows from observing that for all $n\in\{1,\ldots, p-1\}$, $\vartheta_n\le 1$, $\vartheta_p<1$ and
%\begin{equation*}
%\lim_{|z|_\infty\vee|z'|_\infty\to\infty}\frac{J_p((z,z');W)}{W(z,z')}=0\eqsp.
%\end{equation*}
%This completes the proof of \autoref{theo:ergo-convergence:logpois:gen}-- \ref{item:thm:logpois:ergodic:gen:od}. \smallskip \\
%
%{\bf{Proof of \autoref{theo:ergo-convergence:logpois:gen}--\ref{item:thm:logpois:strong:consistency:gen:od}}}. As a consequence, we have ~\ref{assum:gen:identif:unique:pi:gen}. Assumption~\ref{ass:21-lyapunov:gen:od} follows by setting $\bar V=V$.
%Thus, to show the first part of \autoref{theo:ergo-convergence:logpois:gen}--\ref{item:thm:logpois:strong:consistency:gen:od}, we only need to show that Assumptions \ref{assum:continuity:Y:g-theta:gen:od}, \ref{assum:continuity:Y:phi-theta:gen:od} and \ref{assum:practical-cond:O:gen:od} are satisfied. The second part then follows if we can moreover show that \ref{item:ident:G:gen:od} and \ref{item:ident:psi:gen:od} in \autoref{thm:strong:consistency:gen:od} hold.
%
%Clearly, \ref{assum:continuity:Y:g-theta:gen:od} and
%  \ref{assum:continuity:Y:phi-theta:gen:od} hold by the definitions of
%  $\psi^\theta$ and $g^\theta$ given by \eqref{def:psi:logpois:gen:od} and \eqref{eq:density:g:logpois:gen:od}, respectively. It remains to show
%  ~\ref{assum:practical-cond:O:gen:od}. Since $\Theta$ is compact, then
%\begin{equation}\label{eq:set:bound:lpois:consist:gen:od}
% |\omega|\leq\overline{\omega}, \,
%|a_i| \leq\bar a,\,
% \leq |b_i| \leq\bar b,\,
% \bigvee_{{k_1,\ldots,k_p}\in\{1,2,3\}^p}\left\|\prod_{j=1}^p\mathbf{A}_{k_j}\right\|_\infty\le\bar{\rho}, \left\|\mathbf{A}\right\|_\infty\le L,
%\end{equation}
%for some $(\overline{\omega},\bar a,\overline{b},\bar \rho) \in (0,\infty)^3\times(0,1)$ and $L>0$.
%%By \cite[Lemma~12]{moulines-priouret-roueff-2005}, we note that this implies
%%that, for all $\bar\delta\in(\bar\rho,1)$, there exists $\bar C>0$ such that for all
%%$k\geq1$ and all $\theta\in\Theta$,
%%\begin{equation}
%%  \label{eq:uniform-spectral-raius:logpois:gen:od}
%%  \left\|\left(\mathbf{A}+\mathbf{b}\boldsymbol{\gamma}^T\right)^k\right\|\leq
%%  \bar C\,\bar\delta^k\;.
%%\end{equation}
%We set
%$\Zset_1=\Zset$. And so
%\ref{assum:practical-cond:O:gen:od}\ref{assum:exist:practical-cond:subX:gen:od}
%holds. Moreover, for all
%$(\theta,z,u)\in\Theta\times\Zset_1\times\Uset$,
%$g^\theta(\Proj{p}{z};y)\le 1$. Thus,
%Condition~\ref{assum:practical-cond:O:gen:od}\ref{assum:momentCond:gen:od} holds.  Now let
%$z_1\in\Zset$.
%Using ~\eqref{eq:norm-control:logpois:gen:od}, \eqref{eq:psi:logpois:iterate:gen:od} and \eqref{eq:set:bound:lpois:consist:gen:od}, we get that there exist constants $\bar c>0$ and $\bar\delta\in(0,1)$ such that for all $z\in\Zset$, $u_{1:n}\in\Uset^n$ and $\theta\in\Theta$,
%\begin{align*}
%\left|\F{\chunk{u}{1}{n}}(z_1)-\F{\chunk{u}{1}{n}}(z)\right|_\infty&= \left|\mathbf{A}^n(z_1-z)\right|_\infty\\
%&\leq \bar c\,\bar\delta^n\left|z_1-z\right|_\infty\eqsp.
%  \end{align*}
%Thus by \eqref{eq:def:zmet:gen:od}, we get~\ref{assum:practical-cond:O:gen:od}\ref{assum:exist:practical-cond:O:gen:od} with
%$$
%\bar\Psi(z)= c\,\left|z_1-z\right|_\infty\eqsp.
%$$
%Hence~\ref{assum:practical-cond:O:gen:od}\ref{item:psibar:gen:od} holds and since
%$$
%\left|z_1-\Psi^\theta_u(z_1)\right|_\infty\le (L+1)\,|z_1|_\infty+\overline{\omega}+(1\vee\bar b)u,
%$$
%we also get~\ref{assum:practical-cond:O:gen:od}\ref{item:barphi:bar:phi:gen:od} provided that
%\begin{equation}
%  \label{eq:phibar-logpois-temp:gen:od}
%\bar\phi(u)\ge (L+1)\,|z_1|_\infty+\overline{\omega}+(1\vee\bar b)u.
%\end{equation}
%It is straightforward to show that for all $\theta\in\Theta$, $(x,x')\in\Proj{p}{\Zset_1}\times\Proj{p}{\Zset_1}$ and $u\in\Uset$,
%\begin{align*}
%\left|\ln g^\theta\left(x;u\right)-\ln g^\theta\left(x';u\right)\right|&\le |x-x'|\rme^{|x|\vee|x'|}\rme^u \\
%&\le|x-x'|\rme^{\left|x-\Proj{p}{z_1}\right|\vee\left|x'-\Proj{p}{z_1}\right|}\rme^{u+\left|\Proj{p}{z_1}\right|}.
%\end{align*}
%Thus  for all $\theta\in\Theta$, $(z,z')\in\Zset_1\times\Zset_1$ and $u\in\Uset$,
%\begin{align*}
%\left|\ln g^\theta\left(x;u\right)-\ln g^\theta\left(x';u\right)\right|\le|z-z'|_\infty\rme^{|z-z_1|_\infty\vee|z'-z_1|_\infty}\rme^{u+\left|\Proj{p}{z_1}\right|}.
%\end{align*}
%We thus obtain~\ref{assum:practical-cond:O:gen:od}\ref{item:barphi:bar:phi:gen:od},
%\ref{assum:practical-cond:O:gen:od}\ref{item:ineq:loggg:gen:od}
%and~\ref{assum:practical-cond:O:gen:od}\ref{item:H:gen:od} by setting $C=1$,
%$$
%H(s)=s,\quad s\in\rsetp,
%$$
%and
%$$
%\bar\phi(u)=(L+1)\,|z_1|_\infty+\overline{\omega}+(1\vee\bar b)u+\rme^{u+\left|\Proj{p}{z_1}\right|}.
%$$
%In addition, for all $\theta\in\Theta$, $x\in\Xset$ and $\tau>0$, we have
%$$
%\int \rme^{\tau u} G^\theta(x;\rmd u) \le 1+2^\tau\rme^{\tau x} \;.
%$$
%Hence, by letting $\bar V=V$ and using that for all $\theta\in\Theta$, $\pi_1^\theta V<\infty$, with the above definitions, we
%obtain~\ref{assum:practical-cond:O:gen:od}\ref{item:22or23-barphi-V:gen:od}. This establishes the equivalence-class consistency of the \gls{mle} for the log-linear Poisson GARCH$(p,p)$.
%
%
%To conclude the strong consistency of the \gls{mle}, it remains to check \ref{item:ident:G:gen:od} and \ref{item:ident:psi:gen:od} in \autoref{thm:strong:consistency:gen:od}. Condition~\ref{item:ident:G:gen:od} clearly holds by the identifiability of the Poisson distribution. Note that as a consequence of the earlier proof, we have, for all $\theta\in\Theta$, $\tilde\PP^\thv\as$,
%\begin{equation*}%\label{eq:def:psi:stationary:derive:gen:od}
%\F[\theta]{\chunk{U}{-\infty}{0}}=
%\sum_{k=0}^{\infty}
%\mathbf{A}^k (\boldsymbol{\omega}+U_{-k}\mathbf{b})\eqsp.
%\end{equation*}
%By stationarity, for all $\theta\in\Theta$ and $t\in\zset$, we have, $\tilde\PP^\thv\as$,
%\begin{equation*}%\label{eq:def:psi:stationary:derive:gen:od}
%\F[\theta]{\chunk{U}{-\infty}{t}}=
%\sum_{k=0}^{\infty}
%\mathbf{A}^k (\boldsymbol{\omega}+U_{-k+t}\mathbf{b})\eqsp.
%\end{equation*}
%%Now let $\theta=(\omega,\chunk{a}{1}{p},\chunk{b}{1}{p})$ and $\thv=(\omega^\star,\chunk{a^\star}{1}{p},\chunk{b^\star}{1}{p})$. Thus by stationarity, if $\tilde\PP^\thv\as$, $\Proj{p}{\F[\theta]{\chunk{U}{-\infty}{0}}}=\Proj{p}{\F[\thv]{\chunk{U}{-\infty}{0}}}$, then for all $t\in\zset$,
%%\begin{align}\label{eq:stationary:psi:theta:thv:ident:od:gen}
%%\Proj{p}{\F[\theta]{\chunk{U}{-\infty}{t}}}=\Proj{p}{\F[\thv]{\chunk{U}{-\infty}{t}}},\quad\tilde\PP^\thv\as
%%\end{align}
%%Note also that for all $\theta\in\Theta$, we have for all $t\in\zset$,
%%\begin{align*}
%%\Proj{p}{\F[\theta]{\chunk{U}{-\infty}{t}}}=\omega+\sum_{k=0}^pa_k\Proj{p}{\F[\theta]{\chunk{U}{-\infty}{-k-1+t}}}+\sum_{k=0}^pb_kU_{-k+t}
%%\end{align*}
%%It then follows from \eqref{eq:stationary:psi:theta:thv:ident:od:gen} that for all $t\in\zset$, $\tilde\PP^\thv\as$,
%%\begin{align*}
%%&\omega+\sum_{k=0}^pa_k\Proj{p}{\F[\theta]{\chunk{U}{-\infty}{-k-1+t}}}+\sum_{k=0}^pb_kU_{-k+t}\\
%%&=\omega^\star+\sum_{k=0}^pa_k^\star\Proj{p}{\F[\thv]{\chunk{U}{-\infty}{-k-1+t}}}+\sum_{k=0}^pb_k^\star U_{-k+t}.
%%\end{align*}
%%This implies $\tilde\PP^\thv\as$,
%%\begin{align}\label{eq:deduce:ident:theta:thv:od:gen}
%%\begin{split}
%%(b_1^\star-b_1)U_{t} &=(\omega-\omega^\star)+\sum_{k=0}^p(a_k-a_k^\star)\Proj{p}{\F[\thv]{\chunk{U}{-\infty}{-k-1+t}}} \\
%%&+\sum_{k=1}^{p}(b_k-b_k^\star) U_{-k+t}.
%%\end{split}
%%\end{align}
%Moreover, if
%\begin{align*}
%\F[\theta]{\chunk{U}{-\infty}{0}}=\F[\thv]{\chunk{U}{-\infty}{0}},\quad\tilde\PP^\thv\as,
%\end{align*}
%then by stationarity, for all $t\in\zset$,
%\begin{align*}
%\F[\theta]{\chunk{U}{-\infty}{t}}=\F[\thv]{\chunk{U}{-\infty}{t}},\quad\tilde\PP^\thv\as,
%\end{align*}
%and thus
%\begin{equation}\label{eq:ident:obtain:A:k=0:gen:od}
%\sum_{k=0}^{\infty}
%\mathbf{A}^k (\boldsymbol{\omega}+U_{-k+t}\mathbf{b})=\sum_{k=0}^{\infty}\mathbf{A}_\star^k (\boldsymbol{\omega}_\star+U_{-k+t}\mathbf{b}_\star), \quad\tilde\PP^\thv\as
%\end{equation}
%This implies
%\begin{equation}\label{eq:ident:obtain:A:logpois:gen:od}
%(\mathbf{b}-\mathbf{b}_\star)U_{t}=\sum_{k=1}^{\infty}\mathbf{A}_\star^k (\boldsymbol{\omega}_\star+U_{-k+t}\mathbf{b}_\star)-\sum_{k=1}^{\infty}\mathbf{A}^k (\boldsymbol{\omega}+U_{-k+t}\mathbf{b}), \quad\tilde\PP^\thv\as
%\end{equation}
%Conditionally on $\sigma(\chunk{U}{-\infty}{t-1})$, $\left(\rme^{U_t}-1\right)$ is a Poisson variable with positive intensity. Thus the left-hand side of \eqref{eq:ident:obtain:A:logpois:gen:od} is $\tilde\PP^\thv\as$ constant only if $\mathbf{b}=\mathbf{b}_\star$, implying $b_1^\star=b_1$.
%Then, \eqref{eq:ident:obtain:A:k=0:gen:od} reduces to
%\begin{equation}\label{eq:ident:obtain:A:k=1:gen:od}
%\sum_{k=1}^{\infty}
%\mathbf{A}^k (\boldsymbol{\omega}+U_{-k+t}\mathbf{b})=\sum_{k=1}^{\infty}\mathbf{A}_\star^k (\boldsymbol{\omega}_\star+U_{-k+t}\mathbf{b}_\star), \quad\tilde\PP^\thv\as
%\end{equation}
%Thus by repeating the same argument as to obtain $\mathbf{b}=\mathbf{b}_\star$, \eqref{eq:ident:obtain:A:k=1:gen:od} and so on yield that, for all integer $k\ge1$,
%\begin{equation*}%\label{eq:AA:star:b:gen:od}
%(\mathbf{A}^k-\mathbf{A}_\star^k)\mathbf{b}_\star=0.
%\end{equation*}
%And by \eqref{eq:AA:star:b:gen:od}, we have $\mathbf{A}=\mathbf{A}_\star$, yielding that $\left(\chunk{a}{1}{p},\chunk{b}{2}{p}\right)=\left(\chunk{a^\star}{1}{p},\chunk{b^\star}{2}{p}\right)$. It is immediate from \eqref{eq:ident:obtain:A:k=0:gen:od} that $\boldsymbol{\omega}=\boldsymbol{\omega}_\star$, and this is equivalent to having that $\omega=\omega^\star$. The strong consistency therefore follows.
%\smallskip \\
%
%{\bf{Proof of \autoref{theo:ergo-convergence:logpois:gen}--\ref{item:thm:logpois:asymp:normal:gen:od}}}. To conclude the proof, we show that ~\ref{assum:phi:linear:od}--\ref{item:inverse:fisher:gen:od} hold. From the definition of $\Psi^\theta_u(z)$, we have
%$a(\theta,u)=\mathbf{A}$ and $b(\theta,u)=\boldsymbol{\omega}+u\mathbf{b}$, both admitting linear forms in $\theta$. Thus from compactness of $\Theta$ and ~\eqref{eq:def:par:set:logpois:gen} and by noting that for all $\tau>0$,
%\begin{equation}\label{eq:moment:finite:expo:logpois:gen:od}
%\tilde\PE^\thv\left[\rme^{\tau U_1}\right]<\infty\eqsp,
%\end{equation}
%it follows that~\ref{assum:phi:linear:od} holds. Assumptions \ref{assum:differentiability:g:od} and \ref{assum:interior:theta:star} clearly hold. Assumption~\ref{assum:strong:consistent} is immediate as the conclusion of  \autoref{theo:ergo-convergence:logpois:gen}-\ref{item:thm:logpois:strong:consistency:gen:od} above. We have
%\begin{align*}
%\ell_{z,k}^\theta&=\ln g^\theta\left(\Proj{p}{\F{\chunk{U}{1}{k-1}}(z)} ;U_k\right)\\
%			&= (\rme^{U_k}-1)\boldsymbol\delta_p^T\F{\chunk{U}{1}{k-1}}(z)-\rme^{ \boldsymbol\delta_p^T\F{\chunk{U}{1}{k-1}}(z)}-\ln(\rme^{U_k}-1)!,\\
%			\;\\
%\ell_{k}^\theta&=\ln g^\theta\left(\Proj{p}{\F{\chunk{U}{-\infty}{k-1}}(z)} ;U_k\right)\\
%			&=(\rme^{U_k}-1)\boldsymbol\delta_p^T\F{\chunk{U}{-\infty}{k-1}}-\rme^{ \boldsymbol\delta_p^T\F{\chunk{U}{-\infty}{k-1}}}-\ln(\rme^{U_k}-1)!.
%\end{align*}
%Then from \autoref{lem:exist:deriv:psi:od}, for all $i$, $\PPY\as$,
%\begin{align*}
%\frac{\partial\ell_{z,k}^\theta}{\partial\theta_i}&=\left(\rme^{U_k}-1-\rme^{ \boldsymbol\delta_p^T\F{\chunk{U}{1}{k-1}}(z)}\right) \boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{U}{1}{k-1}}(z) \\
%\;\\
%\frac{\partial\ell_{k}^\theta}{\partial\theta_i}&=\left(\rme^{U_k}-1-\rme^{ \boldsymbol\delta_p^T\F{\chunk{U}{-\infty}{k-1}}} \right)\boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{U}{-\infty}{k-1}}
%\end{align*}
%Thus for all $i$, $\PPY\as$,
%
%\begin{align*}
%&\left|\frac{\partial\ell_{z,k}^\theta}{\partial\theta_i}-\frac{\partial\ell_{k}^\theta}{\partial\theta_i}\right|
%%\le
%%\left| \boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{u}{-\infty}{k-1}}- \boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{u}{1}{k-1}}(z)\right|(\rme^{u_k}-1) \\
%%&+\left|\rme^{ \boldsymbol\delta_p^T\F{\chunk{u}{-\infty}{k-1}}} \boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{u}{-\infty}{k-1}}-\rme^{ \boldsymbol\delta_p^T\F{\chunk{u}{1}{k-1}}(z)} \boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{u}{1}{k-1}}(z)\right| (\rme^{u_k}-1)\\
%\le \left| \boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{U}{-\infty}{k-1}}- \boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{U}{1}{k-1}}(z)\right|(\rme^{U_k}-1) \\
%&+\left|\rme^{ \boldsymbol\delta_p^T\F{\chunk{U}{-\infty}{k-1}}}-\rme^{ \boldsymbol\delta_p^T\F{\chunk{U}{1}{k-1}}(z)}\right|\left| \boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{U}{-\infty}{k-1}}\right| \\
%&+\rme^{\left| \boldsymbol\delta_p^T\F{\chunk{U}{1}{k-1}}(z)\right|}\left| \boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{U}{-\infty}{k-1}}- \boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{U}{1}{k-1}}(z)\right| \\
%\end{align*}
%Note that for all $z\in\Zset$ and $k\ge 1$, we have, $\PPY\as$,
%\begin{align*}
%\F{\chunk{U}{-\infty}{k-1}}-\F{\chunk{U}{1}{k-1}}(z)=\mathbf{A}^{k-1}\left(\F{\chunk{U}{-\infty}{0}}-z\right)\eqsp,
%\end{align*}
%and for all $i$,
%\begin{align*}
%\frac{\partial}{\partial\theta_i}\F{\chunk{U}{-\infty}{k-1}}&-\frac{\partial}{\partial\theta_i}\F{\chunk{U}{1}{k-1}}(z) \\
%&=\frac{\partial(\mathbf{A}^{k-1})}{\partial\theta_i}\left(\F{\chunk{U}{-\infty}{0}}-z\right)+\mathbf{A}^{k-1}\frac{\partial}{\partial\theta_i}\F{\chunk{U}{-\infty}{0}}\eqsp.
%\end{align*}
%By similar arguments as in the proof of \autoref{lem:exist:deriv:psi:od} (see~\autoref{sec:proof:lem:deriv:psi:od:gen} in~\autoref{sec:append:proofs:gen:od}) and noting that $\sup_{\theta\in\Theta}\left|\F{\chunk{U}{-\infty}{0}}\right|_\infty$ and $\sup_{\theta\in\Theta}\left|\frac{\partial}{\partial\theta}\F{\chunk{U}{-\infty}{0}}\right|_\infty$ are finite $\tilde\PP^\thv\as$, there exist a constant $\rho_1\in(0,1)$ and a $\tilde\PP^\thv\as$ finite random variable $\tilde M_1=\tilde M_1(z)$ such that for all $k\ge 1$, $\tilde\PP^\thv\as$,
%\begin{align*}
%\left|\F{\chunk{U}{-\infty}{k-1}}-\F{\chunk{U}{1}{k-1}}(z)\right|_\infty\le \tilde M_1\rho_1^k\eqsp,
%\end{align*}
%and for all $i$, $\tilde\PP^\thv\as$,
%\begin{align*}
%\left|\frac{\partial}{\partial\theta_i}\F{\chunk{U}{-\infty}{k-1}}-\frac{\partial}{\partial\theta_i}\F{\chunk{U}{1}{k-1}}(z)\right|_\infty\le \tilde M_1\rho_1^k\eqsp.
%\end{align*}
%%We also have
%%\begin{align}\label{eq:moment:finite:psi:logpois:od:gen}
%%\tilde\PE^\thv\left[\sup_{\theta\in\Theta}\left|\F{\chunk{U}{-\infty}{0}}\right|_\infty\right]<\infty
%%\end{align}
%%and for all $i$
%From  \autoref{lem:exist:deriv:psi:od}, we also have that, for all $i$,
%\begin{align}\label{eq:moment:finite:deriv:psi:logpois:od:gen}
%\tilde\PE^\thv\left[\sup_{\theta\in\Theta}\left|\frac{\partial}{\partial\theta_i}\F{\chunk{U}{-\infty}{0}}\right|_\infty\right]<\infty\eqsp.
%\end{align}
%Recall that for all $t,t'\in\rset$, $\left|\rme^t-\rme^{t'}\right|\le\rme^{|t|\vee |t'|}|t-t'|$.
%Thus, we deduce that
%\begin{align*}
%&\left|\frac{\partial\ell_{z,k}^\theta}{\partial\theta_i}-\frac{\partial\ell_{k}^\theta}{\partial\theta_i}\right| \\
%&\le\rme^{\tilde M_1}\rho_1^k\tilde M_1\left[\rme^{U_k}-1+\rme^{\left|\F{\chunk{U}{-\infty}{k-1}}\right|_\infty}\left(1+\left|\frac{\partial}{\partial\theta_i}\F{\chunk{U}{-\infty}{k-1}}\right|_\infty\right)\right]\eqsp.
%\end{align*}
%On the other hand, $\tilde\PP^\thv\as$, we have
%\begin{align*}
%\rme^{\left|\F{\chunk{U}{-\infty}{0}}\right|_\infty}\le\rme^{\sup_{\theta\in\Theta}\left|\F{\chunk{U}{-\infty}{0}}\right|_\infty}\le\rme^{\sum_{k=0}^\infty\sup_{\theta\in\Theta}\left|\mathbf{A}^k (\boldsymbol{\omega}+U_{-k}\mathbf{b})\right|_\infty}\eqsp.
%\end{align*}
%From ~\eqref{eq:norm-control:logpois:gen:od} and \eqref{eq:set:bound:lpois:consist:gen:od}, there exist constants $c_1, c_2>0$ and $\bar\delta\in(0,1)$ such that, $\tilde\PP^\thv\as$,
%\begin{align*}
%\rme^{\sup_{\theta\in\Theta}\left|\F{\chunk{U}{-\infty}{0}}\right|_\infty}\le c_1\rme^{c_2\sum_{k=0}^\infty\bar\delta^kU_{-k}}\le c_1(1-\bar\delta)\sum_{k=0}^{\infty}\bar\delta^k\rme^{c_2(1-\bar\delta)^{-1}U_{-k}}\eqsp,
%\end{align*}
%and from \eqref{eq:moment:finite:expo:logpois:gen:od}, it follows that for any positive $t>0$,
%\begin{align}\label{eq:finite:moment:exp:phi:lpois:gen:od}
%\tilde\PE^\thv\left[\rme^{t\sup_{\theta\in\Theta}\left|\F{\chunk{U}{-\infty}{0}}\right|_\infty}\right]<\infty\eqsp.
%\end{align}
%Thus, by stationarity and from \eqref{eq:moment:finite:deriv:psi:logpois:od:gen} and \eqref{eq:finite:moment:exp:phi:lpois:gen:od}, \autoref{lem:useful:gen} implies that there exist a constant $\rho_2\in(0,1)$ and a $\tilde\PP^\thv\as$ finite random variable $\tilde M_2=\tilde M_2(z)$ such that for all $i$ and for all $k\ge 1$, $\tilde\PP^\thv\as$,
%\begin{align*}
%&\left|\frac{\partial\ell_{z,k}^\theta}{\partial\theta_i}-\frac{\partial\ell_{k}^\theta}{\partial\theta_i}\right| \le
%\rho_2^k\tilde M_2\eqsp.
%\end{align*}
%Therefore, Assumption~\ref{item:ctl:limit:od:gen} holds. We now check \ref{item:finte:mement:gen:od}. We have
%\begin{align*}
%\ell_{1}^\theta&=\ln g^\theta\left(\Proj{p}{\F{\chunk{U}{-\infty}{0}}} ;U_1\right)\\
%			&= (\rme^{U_1}-1)\boldsymbol\delta_p^T\F{\chunk{U}{-\infty}{0}}-\rme^{ \boldsymbol\delta_p^T\F{\chunk{U}{-\infty}{0}}}-\ln(\rme^{U_1}-1)!.
%\end{align*}
%Then for all $i$ and $j$, we have
%\begin{align*}
%\frac{\partial\ell_{1}^\theta}{\partial\theta_i}	&=\left(\rme^{U_1}-1-\rme^{ \boldsymbol\delta_p^T\F{\chunk{U}{-\infty}{0}}}\right) \boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{U}{-\infty}{0}}
%\end{align*}
%and from \autoref{lem:exist:deriv:psi:od}, $\PPY\as$,
%\begin{align*}
%\frac{\partial^2\ell_{1}^\theta}{\partial\theta_i\partial\theta_j}&=\left(\rme^{U_1}-1-\rme^{ \boldsymbol\delta_p^T\F{\chunk{U}{-\infty}{0}}}\right) \boldsymbol\delta_p^T\frac{\partial^2}{\partial\theta_i\partial\theta_j}\F{\chunk{U}{-\infty}{0}} \\
%&+\rme^{ \boldsymbol\delta_p^T\F{\chunk{U}{-\infty}{0}}}\left( \boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{U}{-\infty}{0}}\right)\left( \boldsymbol\delta_p^T\frac{\partial}{\partial\theta_j}\F{\chunk{U}{-\infty}{0}}\right)\eqsp.
%\end{align*}
%Using similar arguments as the proof of \autoref{lem:exist:deriv:psi:od} together with \eqref{eq:moment:finite:expo:logpois:gen:od}, we can show that for all $i$ and $j$, we have
%\begin{align*}
%\tilde\PE^\thv\left[\sup_{\theta\in\Theta}\left|\frac{\partial}{\partial\theta_i}\F{\chunk{U}{-\infty}{0}}\right|_\infty^2\right]<\infty
%\end{align*}
%and
%\begin{align*}
%\tilde\PE^\thv\left[\sup_{\theta\in\Theta}\left\|\frac{\partial^2}{\partial\theta_i\partial\theta_j}\F{\chunk{U}{-\infty}{0}}\right\|\right]<\infty.
%\end{align*}
%By \eqref{eq:finite:moment:exp:phi:lpois:gen:od} and \eqref{eq:moment:finite:expo:logpois:gen:od}, we can directly show that for all $i$ and $j$,
%\begin{align*}
%\tilde\PE^\thv\left[\sup_{\theta\in\Theta}\left|\frac{\partial\ell_{1}^\theta}{\partial\theta_i}\right|^2\right]<\infty
%\end{align*}
%and
%\begin{align*}
%\tilde\PE^\thv\left[\sup_{\theta\in\Theta}\left\|\frac{\partial^2\ell_{1}^\theta}{\partial\theta_i\partial\theta_j}\right\|\right]<\infty\eqsp.
%\end{align*}
%Thus, \ref{item:finte:mement:gen:od}-\ref{item:1:finte:moment:od:gen} and \ref{item:finte:mement:gen:od}-\ref{item:2:finte:moment:od:gen} hold. Note that for all $i, j$, we have, $\PPY\as$,
%\begin{align*}
%\frac{\partial}{\partial\theta_i}g^\theta\left(\Proj{p}{\F{\chunk{U}{-\infty}{0}}} ;U_1\right)=g^\theta\left(\Proj{p}{\F{\chunk{U}{-\infty}{0}}} ;U_1\right)\frac{\partial\ell_{1}^\theta}{\partial\theta_i}\eqsp.
%\end{align*}
%and
%\begin{align*}
%&\frac{\partial^2}{\partial\theta_i\partial\theta_j}g^\theta\left(\Proj{p}{\F{\chunk{Y}{-\infty}{0}}} ;U_1\right) \\
%&=g^\theta\left(\Proj{p}{\F{\chunk{U}{-\infty}{0}}} ;U_1\right)\frac{\partial^2\ell_{1}^\theta}{\partial\theta_i\partial\theta_j}+\frac{\partial}{\partial\theta_i}g^\theta\left(\Proj{p}{\F{\chunk{U}{-\infty}{0}}} ;Y_1\right)\frac{\partial\ell_{1}^\theta}{\partial\theta_j}
%\end{align*}
%Then, \ref{item:finte:mement:gen:od}-\ref{item:3:finte:moment:od:gen} follows by observing that for all $i$, $\PPY\as$,
%\begin{align*}
%\left|\frac{\partial}{\partial\theta_i}g^\theta\left(\Proj{p}{\F{\chunk{U}{-\infty}{0}}} ;U_1\right)\right|&=g^\theta\left(\Proj{p}{\F{\chunk{U}{-\infty}{0}}} ;U_1\right)\left|\frac{\partial\ell_{1}^\theta}{\partial\theta_i}\right|\le\sup_{\theta\in\Theta}\left|\frac{\partial\ell_{1}^\theta}{\partial\theta_i}\right|\eqsp,
%\end{align*}
%and
%\begin{align*}
%\tilde\PE^\thv\left[\sup_{\theta\in\Theta}\left|\frac{\partial\ell_{1}^\theta}{\partial\theta_i}\right|\right]\le\left(\tilde\PE^\thv\left[\sup_{\theta\in\Theta}\left|\frac{\partial\ell_{1}^\theta}{\partial\theta_i}\right|^2\right]\right)^{1/2}<\infty\eqsp.
%\end{align*}
%Similarly, \ref{item:finte:mement:gen:od}-\ref{item:4:finte:moment:od:gen} holds. To conclude the proof of asymptotic normality, it remains to check \ref{item:inverse:fisher:gen:od}. By \autoref{rem:hessian:invet:gen:od}, it is sufficient to show that \eqref{eq:rem:linear:independent:d-logpois:gen:od} holds. Now let $\boldsymbol{\alpha}=(\alpha_0,\ldots,\alpha_{2p})\in\rset^{2p+1}$ and for convenience, write $\theta=(\theta_0,\ldots,\theta_{2p})=(\omega,\chunk{a}{1:p},\chunk{b}{2}{p},b_1)$. We have, $\PPY\as$,
%\begin{align*}
%\boldsymbol{\alpha}^T\frac{\partial\ell_{1}^\thv}{\partial\theta}=\sum_{i=0}^{2p}\alpha_i\left(\rme^{U_1}-1-\rme^{ \boldsymbol\delta_p^T\F[\thv]{\chunk{U}{-\infty}{0}}}\right) \boldsymbol\delta_p^T\frac{\partial\F[\thv]{\chunk{U}{-\infty}{0}}}{\partial\theta_i}\eqsp.
%\end{align*}
%Since $\left(\rme^{U_1}-1\right)$ conditionally on $\sigma(\chunk{U}{-\infty}{0})$ is a Poisson variable with positive intensity, then $\tilde\PP^\thv\as$, $\boldsymbol{\alpha}^T\frac{\partial\ell_{1}^\thv}{\partial\theta}=0$  implies
%\begin{align}\label{eq:lin:indep:1:gen:od}
%\sum_{i=0}^{2p}\alpha_i \boldsymbol\delta_p^T\frac{\partial\F[\thv]{\chunk{U}{-\infty}{0}}}{\partial\theta_i}=0,\quad\tilde\PP^\thv\as
%\end{align}
%Note that we have
%\begin{align}\label{eq:lin:indep:2:gen:od}
%\frac{\partial\F[\thv]{\chunk{U}{-\infty}{0}}}{\partial\theta_{2p}}=\sum_{k=1}^{\infty}U_{-k}\mathbf{A}_\star^k\boldsymbol{\delta}_p+U_0\boldsymbol{\delta}_p,\quad\tilde\PP^\thv\as
%\end{align}
%Then, since $\left(\rme^{U_0}-1\right)$ conditionally on $\sigma(\chunk{U}{-\infty}{-1})$ is a Poisson variable with positive intensity, we deduce from \eqref{eq:lin:indep:1:gen:od} and \eqref{eq:lin:indep:2:gen:od} that $\alpha_{2p}=0$.
%Following similar argument, we get, for all $k\ge 1$,
%\begin{align*}%\label{eq:invert:A:Astar:zero:gen:od}
%\sum_{i=1}^{2p-1}\alpha_i\boldsymbol{\delta}_p^T\left(\left.\frac{\partial\mathbf{A}^k}{\partial\theta_i}\right|_{\mathbf{A}=\mathbf{A}_\star}\right)\mathbf{b}_\star=0\eqsp.
%\end{align*}
%By \eqref{eq:invert:A:Astar:zero:gen:od}, it yields $\chunk{\alpha}{1}{2p-1}=0$. Then from \eqref{eq:lin:indep:1:gen:od}, we finally have
%\begin{align*}
%\alpha_0\boldsymbol\delta_p^T\frac{\partial}{\partial\theta_0}\F[\thv]{\chunk{U}{-\infty}{0}}=\alpha_0\boldsymbol\delta_p^T\left(\mathbf{I}_{2p-1}-\mathbf{A}_\star\right)^{-1}\boldsymbol\delta_p=0\eqsp,
%\end{align*}
%which implies $\alpha_0=0$ since $\boldsymbol\delta_p^T\left(\mathbf{I}_{2p-1}-\mathbf{A}_\star\right)^{-1}\boldsymbol\delta_p=(1-\sum_{i=1}^pa_i^\star)^{-1}\neq0$.
%\end{proof}



Our next example is the \acrshort{nbingarch}$(p,p)$, $p\in\zsetpnz$, which is defined as follows.
\subsection{NBIN-GARCH$(p,p)$ Model}
\begin{example}[Definition of \acrshort{nbingarch}$(p,p)$]\label{exmpl:nbin-garch:gen} Consider a statistical model recursively defined by: for all $k\in\zsetp$,
\begin{equation}\label{eq:def:nbin-garch:gen}
\begin{split}
&X_{k+1}=\omega+\sum_{i=1}^p{a_i}X_{k-i+1}+\sum_{i=1}^p{b_i}Y_{k-i+1}\eqsp, \\
&Y_{k+1}\mid \mathcal{F}_k\sim \mathcal{NB}\left(r,\frac{X_{k+1}}{1+X_{k+1}}\right)\eqsp,
\end{split}
\end{equation}
where
\begin{itemize}[label=--]
\item
the parameter space
\begin{equation*}%\label{eq:def:theta:set:nbin-garch:gen}
\Theta\subset\{\theta=(\omega,\chunk{a}{1}{p},\chunk{b}{1}{p},r)\in\rsetpnz\times\rsetp^{p}\times\rsetp^{p}\times\rsetpnz\}\eqsp,
\end{equation*} %and $\sum_{i=1}^p(a_i+b_i)> 0$
\item
$\mathcal{F}_k=\sigma\left(\chunk{X}{-p+1}{k+1},\chunk{Y}{-p+1}{k}\right)$,
\item
$\Xset=(0,\infty)$, $\Xsigma=\mathcal{B}(\Xset)$, $\Yset=\zsetp$, $\Ysigma=\mathcal{B}(\Yset)$ and the metrics on $\Xset$ and $\Yset$ are any metrics on $\rset$ and $\zset$, respectively.
\end{itemize}

Denoting for all $(\chunk{x}{1}{p},\chunk{y}{1}{p})\in\Xset^p\times\Yset^p$,
\begin{align}\label{def:phi:nbingarch:gen:od}
\psi_{\chunk{y}{1}{p}}^\theta(\chunk{x}{1}{p})\eqdef\omega+\sum_{i=1}^p{a_i}x_{p-i+1}+\sum_{i=1}^p{b_i}y_{p-i+1}\eqsp,
\end{align}Model~\eqref{eq:def:nbin-garch:gen} can be expressed as the standard form of Model~\eqref{eq:def:nbin-garch:trans:gen} below:
\begin{equation}\label{eq:def:nbin-garch:trans:gen}
\begin{split}
&X_{k+1}=\psi_{\chunk{Y}{k-p+1}{k}}^\theta(\chunk{X}{k-p+1}{k})\eqsp, \\
&Y_{k+1}\mid \mathcal{F}_k\sim \mathcal{NB}\left(r,\frac{X_{k+1}}{1+X_{k+1}}\right)\eqsp.
\end{split}
\end{equation}
The density $g^\theta$ and the probability law $G^\theta$ of $Y_k$ conditional on $X_k=x$ are given by, for all $y\in\Yset$ and $A\in\Ysigma$,
\begin{equation}\label{eq:density:g:pos:gen:nbin}
g^\theta(x;y)=\frac{\Gamma(r+y)}{y\,!\,\Gamma(r)}\left(\frac{1}{1+x}\right)^r\left(\frac{x}{1+x}\right)^y\eqsp,
\end{equation}
and
\begin{equation}\label{eq:density:G:pos:gen:nbin}
G^\theta(x;A)=\sum_{y\in A} \1_{A}(y)g^\theta(x;y)\eqsp.
\end{equation}
According to \autoref{def:obs-driv-gen}, Model~\eqref{eq:def:nbin-garch:gen} is an \acrshort{odm}$(p,p)$ and is dominated by the counting measure on $\zset$. Now, let $\Zset=\Xset^p\times\Yset^{p-1}$ and $\Zsigma=\Xsigma^{\otimes p}\otimes\Ysigma^{\otimes {(p-1)}}$ and define the measurable function $\Psi^\theta$ similarly as in~\eqref{eq:def:psi:gen:od}, which is given by, for all $(z,y)\in\Zset\times\Yset$,
\begin{equation}\label{eq:psi:nbin-garch:gen}
\Psi_{y}^\theta(z)=\boldsymbol{\omega}+\mathbf{A}z+y\mathbf{b}\eqsp.
\end{equation}

The process $\{Z_k: k\in\zsetp\}$, where $Z_k=(\chunk{X}{k-p+1}{k},\chunk{Y}{k-p+1}{k-1})$, is a Markov chain on $(\Zset,\Zsigma)$ with transition kernel $R^\theta$ given by \eqref{eq:def:kernel:R:gen} with $\Psi^\theta$ and $G^\theta$ defined in  \eqref{eq:density:G:pos:gen:nbin} and \eqref{eq:psi:nbin-garch:gen}, respectively.
Let us now state our ergodicity, consistency and asymptotic normality results for the \acrshort{nbingarch}$(p,p)$ model.
\end{example}


 \begin{proposition}\label{theo:ergo-convergence:garch:gen:nbin}
  Suppose that $\theta\in\Theta$ satisfies

  \begin{equation}\label{eq:con:ergo:gen:nbin}
  \radius(\mathbf{A}+r\mathbf{b}\boldsymbol{\delta}_p^T)<1\eqsp.
  \end{equation}
  \begin{enumerate}[label=(\roman*)]
  \item\label{item:thm:nbin:ergo:gen:od}
   Then the conclusion of \autoref{thm:ergodicity:gen} holds with $\bar V(z)=\mathbf{1}^Tz$ for all $z\in\Zset$.
   \item\label{item:thm:nbin:strong:consistency:gen:od}
\autoref{thm:convergence-main:gen:od} holds with any $z_1\in\Zset$. If the true parameter $\thv=(\omega^\star,\chunk{a^\star}{1}{p},\chunk{b^\star}{1}{p})$ moreover satisfies: for all $k\in\zsetpnz$,
\begin{equation}\label{eq:AA:star:b:gen:od:nbin}
(\mathbf{A}^k-\mathbf{A}_\star^k)\mathbf{b}_\star=0\quad \text{implies} \quad\mathbf{A}=\mathbf{A}_\star\eqsp,
\end{equation}
then \autoref{thm:strong:consistency:gen:od} also holds.
\end{enumerate}
\end{proposition}

\begin{remark}
From \autoref{lem:spec:rad:pos:1}, Condition~\eqref{eq:con:ergo:gen:nbin} is equivalent to $\sum_{i=1}^p(a_i+rb_i)<1$.
\end{remark}

\begin{proof} {\textbf{Proof of \autoref{theo:ergo-convergence:garch:gen:nbin}--\ref{item:thm:nbin:ergo:gen:od}.}} In this part, for convenience, let us drop the super script $\theta$ from for example $G^\theta$, $R^\theta$, etc. Now set $V=\bar V$. From \autoref{thm:ergodicity:gen}, we need to show that ~\ref{ass-ergo-Zset:gen}, \ref{assum:weakFeller-V:gen},  \ref{assum:reachable:gen}, \ref{assum:bound:rho:gen}
  and~\ref{assum:alpha-phi:gen} hold. Assumption~~\ref{ass-ergo-Zset:gen} holds with any metric $\Zmet$
associated to a norm on the finite dimensional space $\Zset$ so that $\Zsigma$ is the associated Borel $\sigma$-field (The precise choice of this metric is
postponed to the verification of~\ref{assum:bound:rho:gen}).
The fact that $R$ is weak Feller follows by observing that for fixed $r>0$, as $p\to p'$, $\mathcal{NB}(r,p)$ converges weakly to $\mathcal{NB}(r,p')$. Moreover, we have, for all $z\in\Zset$,
\begin{align*}
RV(z)
&=\int V(\boldsymbol{\omega}+\mathbf{A}z+y\mathbf{b})\;G(\Proj{p}{z};\rmd y)\\
&=
\mathbf{1}^T\boldsymbol{\omega}+\mathbf{1}^T\left(\mathbf{A}+r\mathbf{b}\boldsymbol{\delta}_p^T\right)z\eqsp.
\end{align*}
By iteration, we obtain that for all $k\in\zsetpnz$,
\begin{align*}
R^kV(z)=\mathbf{1}^T\boldsymbol{\omega}+\cdots+\mathbf{1}^T\left(\mathbf{A}+r\mathbf{b}\boldsymbol{\delta}_p^T\right)^{k-1}\boldsymbol{\omega}+\mathbf{1}^T\left(\mathbf{A}+r\mathbf{b}\boldsymbol{\delta}_p^T\right)^kz\eqsp.
\end{align*}
Since $\radius(\mathbf{A}+r\mathbf{b}\boldsymbol{\delta}_p^T)<1$, then there exists a pair $(q,\rho)\in\zsetpnz\times(0,1)$ such that
\begin{align*}
R^qV(z)\le\mathbf{1}^T\boldsymbol{\omega}+\cdots+\mathbf{1}^T\left(\mathbf{A}+r\mathbf{b}\boldsymbol{\delta}_p^T\right)^{q-1}\boldsymbol{\omega}+\rho^q\mathbf{1}^Tz\eqsp.
\end{align*}

Hence~\ref{assum:weakFeller-V:gen}
holds with some $q\ge1$, $\lambda=\rho^q<1$ and $\beta=\mathbf{1}^T\boldsymbol{\omega}+\cdots+\mathbf{1}^T\left(\mathbf{A}+r\mathbf{b}\boldsymbol{\delta}_p^T\right)^{q-1}\boldsymbol{\omega}\ge0$. We now show \ref{assum:reachable:gen}.
Let us use the norm
$$
\|\mathbf{M}\|= \bigvee_{j=1}^{2p-1} \sum_{i=1}^{2p-1} |\mathbf{M}_{i,j}|=\sup_{|z|\leq1}|\mathbf{M}z|
$$
on $(2p-1)\times (2p-1)$ matrices, where $|z|=\sum_{i=1}^{2p-1}|z_i|$ is a norm of $z=\chunk{z}{1}{2p-1}\in\Zset$.
Note that
from~\eqref{eq:con:ergo:gen:nbin}, there exists $\delta\in(0,1)$ and $c>0$ such
that, for any $k\geq1$,
\begin{equation}
  \label{eq:norm-control:gen:nbin}
\left\|\mathbf{A}^k\right\|\leq c\;
\delta^k \;.
\end{equation}
Hence $(\mathbf{I}_{2p-1}-\mathbf{A})^{-1}=\mathbf{I}_{2p-1}+\sum_{k\ge1}\mathbf{A}^k$ is well defined and we set
$z_\infty=(\mathbf{I}_{2p-1}-\mathbf{A})^{-1}\boldsymbol{\omega}$. Let $\mathcal{N}_\infty$ be an open neighborhood of $z_\infty$ and let $z\in\Zset$. Define recursively the sequence $z_0 = z$ and for all $k\ge1$, $z_k=\boldsymbol{\omega}+\mathbf{A}z_{k-1}$. From~\eqref{eq:norm-control:gen:nbin}, this so-defined sequence admits a unique limiting point $\lim_{n\to\infty}z_n=z_\infty$. Thus, there exists some $n\in\zsetpnz$ such that for all $k\ge n$, $z_k\in\mathcal{N}_\infty$. For such $n$, we have
\begin{align*}
R^n(z,\mathcal{N}_\infty)=\bar\PP_{z}(Z_n\in\mathcal{N}_\infty)&\ge\bar\PP_{z}(Z_k=z_k \,\, \text{for all} \,\, k=1,\ldots,n)\\
&=\bar\PP_{z}(Y_0=\cdots=Y_{n-1}=0)>0\eqsp.
\end{align*}

Next we prove~\ref{assum:bound:rho:gen}. We have
$$
\Psi_y(z)-\Psi_y(z')=\mathbf{A}(z-z') \;.
$$
Since~\eqref{eq:con:ergo:gen:nbin} implies that
$\radius(\mathbf{A})<1$, there exists a vector norm $|\cdot|_\Zset$, which makes $\mathbf{A}$
strictly contracting. Choosing the metric $\Zmet$ on $\Zset$ (embedded in $\rset^{2p-1}$) as the one
derived from this norm, we get~\ref{assum:bound:rho:gen}. To
show~\ref{assum:alpha-phi:gen}, we rely
on~\autoref{lem:det:alpha:gen:od}. Let us set $\Cset=(0,\infty)=\Xset$ and
$\Sset=\{1\}$, then the probability measure $\mu$ boils down the Dirac mass on $\Sset$. For all $(x,y)\in\Xset\times\Yset$, let $j(x)=(1+x)^{-r}$ and
$h(x;y)=\frac{\Gamma(r+y)}{y\,!\,\Gamma(r)}\left(\frac{x}{1+x}\right)^y$. Indeed,
$h$ satisfies ~\ref{item:F1:gen}. Hence, from lemma~\autoref{lem:det:alpha:gen:od} we
$\alpha$ and $\phi$ are obtained by setting: for all $z,  z'\in\Zset$,
$$\alpha(z,z')=\left(\frac{1+\Pi_{p}(z)\wedge\Pi_{p}(z')}{1+\Pi_{p}(z)\vee\Pi_{p}(z')}\right)^r
$$
and
$$
\phi(z,z')=\Pi_{p}(z)\wedge\Pi_{p}(z')\eqsp.
$$

These so-defined functions $\alpha$ and $\phi$ indeed satisfy ~\ref{assum:alpha-phi:gen}\ref{item:alpha-phi:3:gen}. For any given $r>0$, for all $(z, z')\in\Zset^2$, we have
\begin{align*}
1-\alpha(z,z') &=1-\left(\frac{1+\Proj{p}{z}\wedge \Proj{p}{z'}}{1+\Proj{p}{z}\vee \Proj{p}{z'}}\right)^r \\
&\leq (1\vee r)\left|\Proj{p}{z}-\Proj{p}{z'}\right|\\
& \leq W(z,z')\;\Zmet(z,z') \;,
\end{align*}
where $\Zmet$ is the metric previously defined and $W$ is defined by
$W(z,z')=c_{\Zmet}(1\vee r)$ with $c_{\Zmet}>0$ is
conveniently chosen (such a constant exists since $\Zmet$ is the metric
associated to a norm and $\Zset$ is a subset of finite dimensional space).
Thus,
~\ref{assum:alpha-phi:gen}\ref{assum:definition-gamma-x:gen} and
~\ref{assum:alpha-phi:gen}\ref{assum:hyp:1-alpha:W:gen} hold and, since for all
$y\in\Yset$ and $z\in\Zset$, $\Proj{p}{\Psi_y(z)}\ge\omega$,
$W(\Psi_y(z),\Psi_y(z'))$ is uniformly bounded over
$(z,z', y)\in\Zset\times\Zset\times\Yset$
and~\ref{assum:alpha-phi:gen}\ref{assum:driftCond:W:new:gen:od} holds. This
completes the proof. %\textbf{Step 1}.....
\smallskip \\

{\bf{Proof of  \autoref{theo:ergo-convergence:garch:gen:nbin}--\ref{item:thm:nbin:strong:consistency:gen:od}}}. Note that \ref{ass:21-lyapunov:gen:od} immediately follows. Thus, to show the first part of \autoref{theo:ergo-convergence:logpois:gen}--\ref{item:thm:logpois:strong:consistency:gen:od}, we only need to show that Assumptions \ref{assum:continuity:Y:g-theta:gen:od}, \ref{assum:continuity:Y:phi-theta:gen:od} and \ref{assum:practical-cond:O:gen:od} are satisfied. The second part then follows if we can moreover show that \ref{item:ident:G:gen:od} and \ref{item:ident:psi:gen:od} in \autoref{thm:strong:consistency:gen:od} hold.

Clearly, \ref{assum:continuity:Y:g-theta:gen:od} and
  \ref{assum:continuity:Y:phi-theta:gen:od} hold by the definitions of
  $\psi^\theta$ and $g^\theta$ given by \eqref{def:phi:nbingarch:gen:od} and \eqref{eq:density:g:pos:gen:nbin}, respectively. It remains to show
  ~\ref{assum:practical-cond:O:gen:od}. Since $\Theta$ is compact, then
\begin{equation}\label{eq:set:bound:nbin:consist:gen:od}
 \underline{\omega}\le\omega\leq\overline{\omega}, \,
0\le a_i \leq\bar a,\,
 0\leq b_i \leq\bar b,\,
\radius(\mathbf{A}+r\mathbf{b}\boldsymbol{\delta}_p^T)\le\bar{\rho}, \, \underline{r}\le r\le \overline{r}
\end{equation}
for some $(\underline{\omega},\overline{\omega},\bar a,\overline{b},\underline{r},\overline{r},\bar \rho) \in (0,\infty)^6\times(0,1)$.
%By \cite[Lemma~12]{moulines-priouret-roueff-2005}, we note that this implies
%that, for all $\bar\delta\in(\bar\rho,1)$, there exists $\bar C>0$ such that for all
%$k\geq1$ and all $\theta\in\Theta$,
%\begin{equation}
%  \label{eq:uniform-spectral-raius:logpois:gen:od}
%  \left\|\left(\mathbf{A}+\mathbf{b}\boldsymbol{\gamma}^T\right)^k\right\|\leq
%  \bar C\,\bar\delta^k\;.
%\end{equation}

We set
$\Zset_1=\Zset$ so that
\ref{assum:practical-cond:O:gen:od}\ref{assum:exist:practical-cond:subX:gen:od}
holds. Moreover, for all
$(\theta,z,y)\in\Theta\times\Zset_1\times\Yset$,
$g^\theta(\Proj{p}{z};y)\le 1$. Thus,
Condition~\ref{assum:practical-cond:O:gen:od}\ref{assum:momentCond:gen:od} holds.  Note that we have, for all $n\in\zsetpnz$ and $z\in\Zset$,
\begin{equation}\label{eq:psi:nbingach:iterate:gen:od}
\F[\theta]{\chunk{y}{0}{n-1}}(z)=
\mathbf{A}^{n}z+\sum_{j=0}^{n-1}
\mathbf{A}^j (\boldsymbol{\omega}+y_{n-1-j}\mathbf{b})\eqsp.
\end{equation}
Now let $|\cdot|$ be the Taxicab norm on $\Zset$ and let $z_1\in\Zset$. Using ~\eqref{eq:norm-control:gen:nbin}, \eqref{eq:set:bound:nbin:consist:gen:od} and \eqref{eq:psi:nbingach:iterate:gen:od}, we get that there exist constants $\bar c>0$ and $\bar\delta\in(0,1)$ such that for all $z\in\Zset$, $y_{1:n}\in\Yset^n$ and $\theta\in\Theta$,
\begin{align*}
\Zmet\left(\F{\chunk{y}{1}{n}}(z_1),\F{\chunk{y}{1}{n}}(z)\right)&= \left|\mathbf{A}^n(z_1-z)\right|_\Zset\\
&\leq \bar c\,\bar\delta^n\left|z_1-z\right|\eqsp.
  \end{align*}
Thus, we get~\ref{assum:practical-cond:O:gen:od}\ref{assum:exist:practical-cond:O:gen:od} with
$$
\bar\Psi(z)= \bar c\left|z_1-z\right|\eqsp.
$$
Hence~\ref{assum:practical-cond:O:gen:od}\ref{item:psibar:gen:od} holds and since
$$
\left|z_1-\Psi^\theta_y(z_1)\right|\le (L+1)\,|z_1|+\overline{\omega}+(1+\bar b)y,
$$
for some $L>0$, we also get~\ref{assum:practical-cond:O:gen:od}\ref{item:barphi:bar:phi:gen:od} provided that for all $y\in\Yset$,
\begin{equation}
  \label{eq:phibar-nbin-temp:gen:od}
\bar\phi(y)\ge (L+1)|z_1|+\overline{\omega}+(1+\bar b)y.
\end{equation}
For all $\theta\in\Theta$, $(x,x')\in\Proj{p}{\Zset_1}\times\Proj{p}{\Zset_1}$ and $y\in\Yset$,
\begin{align*}
\left|\ln g^\theta (x;y)-\ln g^\theta (x';y)\right|
&=\left|(r+y)[\ln(1+x')-\ln(1+x)]+y[\ln x-\ln x']\right|\\
&\leq \left[(r+y)(1+\underline \omega)^{-1}+y\,\underline
  \omega^{-1}\right]\;|x-x'|\\
&\leq \left[\overline r +y\,(1+\underline
  \omega^{-1})\right]\;|x-x'| \;.
\end{align*}
We thus obtain~\ref{assum:practical-cond:O:gen:od}\ref{item:barphi:bar:phi:gen:od},
\ref{assum:practical-cond:O:gen:od}\ref{item:ineq:loggg:gen:od}
and~\ref{assum:practical-cond:O:gen:od}\ref{item:H:gen:od} by setting $C=0$, $H(s)=s$
and $\bar\phi(y)=\bar\omega\vee{\bar r}+(1+L)|z_1|+\left(\bar{b}+1+\underline \omega^{-1}\right)y$.
Observe that for all $z\in\Zset$,
$$
\int \lnp y \; G^\theta(\Proj{p}{z};\rmd y) \leq \int y \; G^\theta(\Proj{p}{z};\rmd y) = r \Proj{p}{z}\leq
\overline r \bar V(z)\eqsp.
$$
Hence, using that for all $\theta\in\Theta$, $\pi_1^\theta V<\infty$,  we
obtain~\ref{assum:practical-cond:O:gen:od}\ref{item:22or23-barphi-V:gen:od}, establishing the equivalence-class consistency of the \gls{mle} for the \acrshort{nbingarch}$(p,p)$ model.


To conclude the strong consistency of the \gls{mle}, it remains to check \ref{item:ident:G:gen:od} and \ref{item:ident:psi:gen:od} in \autoref{thm:strong:consistency:gen:od}. Condition~\ref{item:ident:G:gen:od} clearly holds by the identifiability of the negative binomial distribution. Note that as a consequence of the earlier proof, we have, for all $\theta\in\Theta$, $\tilde\PP^\thv\as$,
\begin{equation*}%\label{eq:def:psi:stationary:derive:gen:od}
\F[\theta]{\chunk{Y}{-\infty}{0}}=
\sum_{k=0}^{\infty}
\mathbf{A}^k (\boldsymbol{\omega}+Y_{-k}\mathbf{b})\eqsp.
\end{equation*}
By stationarity, for all $\theta\in\Theta$ and $t\in\zset$, we have, $\tilde\PP^\thv\as$,
\begin{equation*}%\label{eq:def:psi:stationary:derive:gen:od}
\F[\theta]{\chunk{Y}{-\infty}{t}}=
\sum_{k=0}^{\infty}
\mathbf{A}^k (\boldsymbol{\omega}+Y_{-k+t}\mathbf{b})\eqsp.
\end{equation*}
%Now let $\theta=(\omega,\chunk{a}{1}{p},\chunk{b}{1}{p})$ and $\thv=(\omega^\star,\chunk{a^\star}{1}{p},\chunk{b^\star}{1}{p})$. Thus by stationarity, if $\tilde\PP^\thv\as$, $\Proj{p}{\F[\theta]{\chunk{U}{-\infty}{0}}}=\Proj{p}{\F[\thv]{\chunk{U}{-\infty}{0}}}$, then for all $t\in\zset$,
%\begin{align}\label{eq:stationary:psi:theta:thv:ident:od:gen}
%\Proj{p}{\F[\theta]{\chunk{U}{-\infty}{t}}}=\Proj{p}{\F[\thv]{\chunk{U}{-\infty}{t}}},\quad\tilde\PP^\thv\as
%\end{align}
%Note also that for all $\theta\in\Theta$, we have for all $t\in\zset$,
%\begin{align*}
%\Proj{p}{\F[\theta]{\chunk{U}{-\infty}{t}}}=\omega+\sum_{k=0}^pa_k\Proj{p}{\F[\theta]{\chunk{U}{-\infty}{-k-1+t}}}+\sum_{k=0}^pb_kU_{-k+t}
%\end{align*}
%It then follows from \eqref{eq:stationary:psi:theta:thv:ident:od:gen} that for all $t\in\zset$, $\tilde\PP^\thv\as$,
%\begin{align*}
%&\omega+\sum_{k=0}^pa_k\Proj{p}{\F[\theta]{\chunk{U}{-\infty}{-k-1+t}}}+\sum_{k=0}^pb_kU_{-k+t}\\
%&=\omega^\star+\sum_{k=0}^pa_k^\star\Proj{p}{\F[\thv]{\chunk{U}{-\infty}{-k-1+t}}}+\sum_{k=0}^pb_k^\star U_{-k+t}.
%\end{align*}
%This implies $\tilde\PP^\thv\as$,
%\begin{align}\label{eq:deduce:ident:theta:thv:od:gen}
%\begin{split}
%(b_1^\star-b_1)U_{t} &=(\omega-\omega^\star)+\sum_{k=0}^p(a_k-a_k^\star)\Proj{p}{\F[\thv]{\chunk{U}{-\infty}{-k-1+t}}} \\
%&+\sum_{k=1}^{p}(b_k-b_k^\star) U_{-k+t}.
%\end{split}
%\end{align}
Moreover, if
\begin{align*}
\F[\theta]{\chunk{Y}{-\infty}{0}}=\F[\thv]{\chunk{Y}{-\infty}{0}},\quad\tilde\PP^\thv\as,
\end{align*}
then by stationarity, for all $t\in\zset$,
\begin{align*}
\F[\theta]{\chunk{Y}{-\infty}{t}}=\F[\thv]{\chunk{Y}{-\infty}{t}},\quad\tilde\PP^\thv\as,
\end{align*}
and thus
\begin{equation}\label{eq:ident:obtain:A:k=0:gen:od:nbin}
\sum_{k=0}^{\infty}
\mathbf{A}^k (\boldsymbol{\omega}+Y_{-k+t}\mathbf{b})=\sum_{k=0}^{\infty}\mathbf{A}_\star^k (\boldsymbol{\omega}_\star+Y_{-k+t}\mathbf{b}_\star), \quad\tilde\PP^\thv\as
\end{equation}
This implies
\begin{equation}\label{eq:ident:obtain:A:nbin:gen:od}
(\mathbf{b}-\mathbf{b}_\star)Y_{t}=\sum_{k=1}^{\infty}\mathbf{A}_\star^k (\boldsymbol{\omega}_\star+Y_{-k+t}\mathbf{b}_\star)-\sum_{k=1}^{\infty}\mathbf{A}^k (\boldsymbol{\omega}+Y_{-k+t}\mathbf{b}), \quad\tilde\PP^\thv\as
\end{equation}
Conditionally on $\sigma(\chunk{Y}{-\infty}{t-1})$, $Y_t$ is negative binomial variable with parameter $(r_\star,p_\star)\in(0,\infty)\times(0,1)$. Thus, the left-hand side of \eqref{eq:ident:obtain:A:nbin:gen:od} is $\tilde\PP^\thv\as$ constant only if $\mathbf{b}=\mathbf{b}_\star$, implying $b_1^\star=b_1$.
Then, \eqref{eq:ident:obtain:A:k=0:gen:od:nbin} reduces to
\begin{equation}\label{eq:ident:obtain:A:k=1:gen:od:nbin}
\sum_{k=1}^{\infty}
\mathbf{A}^k (\boldsymbol{\omega}+Y_{-k+t}\mathbf{b})=\sum_{k=1}^{\infty}\mathbf{A}_\star^k (\boldsymbol{\omega}_\star+Y_{-k+t}\mathbf{b}_\star), \quad\tilde\PP^\thv\as
\end{equation}
Thus, by repeating the same argument as to obtain $\mathbf{b}=\mathbf{b}_\star$, \eqref{eq:ident:obtain:A:k=1:gen:od:nbin} and so on yield that, for all integer $k\ge1$,
\begin{equation*}%\label{eq:AA:star:b:gen:od}
(\mathbf{A}^k-\mathbf{A}_\star^k)\mathbf{b}_\star=0.
\end{equation*}
And by \eqref{eq:AA:star:b:gen:od:nbin}, we have $\mathbf{A}=\mathbf{A}_\star$, yielding that $\left(\chunk{a}{1}{p},\chunk{b}{2}{p}\right)=\left(\chunk{a^\star}{1}{p},\chunk{b^\star}{2}{p}\right)$. It is immediate from \eqref{eq:ident:obtain:A:k=0:gen:od:nbin} that $\boldsymbol{\omega}=\boldsymbol{\omega}_\star$, and this is equivalent to having that $\omega=\omega^\star$. Thus, \ref{item:ident:psi:gen:od} is satisfied and the strong consistency therefore follows.
\end{proof}

\begin{proposition}\label{thm:nbin:asymp:normal:gen:od}
  Suppose that $\theta\in\Theta$ satisfies
  \begin{equation}\label{eq:con:normal:gen:nbin}
  \radius\left(\mathbf{A}^T\mathbf{A}+r\left(\mathbf{A}^T\mathbf{b}\boldsymbol{\delta}_p^T+\boldsymbol{\delta}_p\mathbf{b}^T\mathbf{A}\right)+r(r+1)\boldsymbol{\delta}_p\mathbf{b}^T\mathbf{b}\boldsymbol{\delta}_p^T\right)<1\eqsp.
  \end{equation}
In addition, assume that the true parameter $\thv$ lies in the interior of $\Theta$ and satisfies  \eqref{eq:AA:star:b:gen:od:nbin},
% and, for all $k\in\zsetpnz$ and $\chunk{\alpha}{1}{2p-1}\in\rset^{2p-1}$, writing $\chunk{\theta}{1}{2p-1}=(\chunk{a}{1}{p},\chunk{b}{2}{p})$,
%\begin{align}\label{eq:invert:A:Astar:zero:gen:od:nbin}
%\sum_{i=1}^{2p-1}\alpha_i\boldsymbol{\delta}_p^T\left(\left.\frac{\partial\mathbf{A}^k}{\partial\theta_i}\right|_{\mathbf{A}=\mathbf{A}_\star}\right)\mathbf{b}_\star=0 \quad \implies \quad \chunk{\alpha}{1}{2p-1}=0\eqsp,
%\end{align}
  then \autoref{thm:asym:normal:od:gen} holds.
\end{proposition}

\begin{remark}\label{rem:strong;consist:nbin:gen:od}
Condition~\eqref{eq:con:normal:gen:nbin} implies Condition~\eqref{eq:con:ergo:gen:nbin}. Thus the stationarity and ergodicity for the model follow. If moreover \eqref{eq:AA:star:b:gen:od:nbin} is satisfied, then the strong consistency, that is, $\lim_{n\to\infty}\mlY{z,n}=\thv$, $\PPY\as$, holds for any $z\in\Zset$.
\end{remark}
Before proving \autoref{thm:nbin:asymp:normal:gen:od}, let us show the following result.
\begin{lemma}
Assume \eqref{eq:con:normal:gen:nbin}. Then
\begin{align}\label{eq:finite:second:mement:Y:nbin:gen:od}
\PEY[Y_1^2]<\infty.
\end{align}
\end{lemma}

\begin{proof}
Now, assume that \eqref{eq:con:normal:gen:nbin} holds. To obtain \eqref{eq:finite:second:mement:Y:nbin:gen:od} it is sufficient to show that there exists a triplet $(q,\lambda,\beta)\in\zsetp\times(0,1)\times\rsetp$ such that $R^q\tilde{V}<\lambda \tilde{V}+\beta$ with $\tilde V(z)=z^Tz$. Note first that for $U\sim \mathcal{NB}(r,x/(1+x))$, we have $\PE[U]=rx$ and $\PE[U^2]=r(r+1)x^2+rx$. Note also that the matrix $\mathbf{M}\eqdef\mathbf{A}^T\mathbf{A}+r\left(\mathbf{A}^T\mathbf{b}\boldsymbol{\delta}_p^T+\boldsymbol{\delta}_p\mathbf{b}^T\mathbf{A}\right)+r(r+1)\boldsymbol{\delta}_p\mathbf{b}^T\mathbf{b}\boldsymbol{\delta}_p^T$ is symmetric positive semidefinite.
%Thus by Cholesky decomposition, the exists a lower triangular matrix $\boldsymbol{\Lambda}$ such that $\mathbf{M}=\boldsymbol{\Lambda}\boldsymbol{\Lambda}^T$.
Thus, there exist a diagonal matrix $\mathbf{D}\ge 0$ and an orthogonal matrix $\mathbf{Q}$ such that $\mathbf{M}=\mathbf{Q}\mathbf{D}\mathbf{Q}^T$; and, moreover,  $\radius(\mathbf{M})=\radius(\mathbf{D})$.

For all $z\in\Zset$, we have
\begin{align*}
R\tilde{V}(z)
&=\int \tilde{V}(\boldsymbol{\omega}+\mathbf{A}z+y\mathbf{b})\,G(\Proj{p}{z};\rmd y)\\
&=\int ({\boldsymbol\omega}+\mathbf{A}z+y\mathbf{b})^T({\boldsymbol\omega}+\mathbf{A}z+y\mathbf{b})\,G(\boldsymbol{\delta}_p^Tz;\rmd y)\\
&=
L_1(z)+z^T\mathbf{M}z=L_1(z)+z^T\mathbf{Q}\mathbf{D}\mathbf{Q}^Tz\eqsp \\
&\le L_1(z)+\radius(\mathbf{D})z^T\mathbf{Q}\mathbf{Q}^Tz=L_1(z)+\radius(\mathbf{M})\tilde{V}(z),
\end{align*}
where $L_1(z)={\boldsymbol\omega}^T{\boldsymbol\omega}+\left(2{\boldsymbol\omega}^T\mathbf{A}+r2{\boldsymbol\omega}^T\mathbf{b}\boldsymbol{\delta}_p^T+r\mathbf{b}^T\mathbf{b}\boldsymbol{\delta}_p^T\right)z$, which is a linear function in $z$. Since $\radius(\mathbf{M})<1$, and
$$
\lim_{|z|\to\infty}\frac{R\tilde{V}(z)}{\tilde{V}(z)}\le\radius(\mathbf{M})<1,
$$
then the exists $(\beta,\lambda)\in\rsetp\times (0,1)$ such that for all $z\in\Zset$,
\begin{align*}
R\tilde{V}(z)\le \lambda \tilde{V}(z)+\beta\eqsp,
\end{align*}
completing the proof.
\end{proof}

\begin{proof}[Proof of \autoref{thm:nbin:asymp:normal:gen:od}]
%{\textbf{Proof of \autoref{thm:nbin:asymp:normal:gen:od}.}}
Assumption~\ref{assum:gen:identif:unique:pi:gen} holds by \autoref{rem:strong;consist:nbin:gen:od}. We now show that ~\ref{assum:phi:linear:od}--\ref{item:inverse:fisher:gen:od} are satisfied. From the definition of $\Psi^\theta_y(z)$, we have
$a(\theta,y)=\mathbf{A}$ and $b(\theta,y)=\boldsymbol{\omega}+y\mathbf{b}$, both admitting linear forms in $\theta$. From compactness of $\Theta$ and by noting that ~\eqref{eq:con:normal:gen:nbin} implies \ref{assum:phi:linear:od}\ref{item:ineq:prod:rho:n:od} and that \eqref{eq:finite:second:mement:Y:nbin:gen:od} also implies
\begin{equation}\label{eq:moment:finite:nbin:gen:od}
\tilde\PE^\thv\left[Y_1\right]<\infty\eqsp,
\end{equation}
it follows that~\ref{assum:phi:linear:od} holds. Assumptions \ref{assum:differentiability:g:od} and \ref{assum:interior:theta:star} are clearly satisfied. Assumption~\ref{assum:strong:consistent} immediately follows by \autoref{rem:strong;consist:nbin:gen:od}. We have
\begin{align*}
\ell_{z,k}^\theta&=\ln g^\theta\left(\Proj{p}{\F{\chunk{Y}{1}{k-1}}(z)} ;Y_k\right)\\
			&= \ln\Gamma(Y_k+r)-(Y_k+r)\ln\left(1+\boldsymbol\delta_p^T\F{\chunk{Y}{1}{k-1}}(z)\right) \\
			&+Y_k\ln\left(\boldsymbol\delta_p^T\F{\chunk{Y}{1}{k-1}}(z)\right)-\ln\Gamma(r)-\ln(Y_k!),\\
			\;\\
\ell_{k}^\theta&=\ln g^\theta\left(\Proj{p}{\F{\chunk{Y}{-\infty}{k-1}}} ;Y_k\right)\\
				&= \ln\Gamma(Y_k+r)-(Y_k+r)\ln\left(1+\boldsymbol\delta_p^T\F{\chunk{Y}{-\infty}{k-1}}\right) \\
			&+Y_k\ln\left(\boldsymbol\delta_p^T\F{\chunk{Y}{-\infty}{k-1}}\right)-\ln\Gamma(r)-\ln(Y_k!).
\end{align*}
Then from \autoref{lem:exist:deriv:psi:od}, for all $\theta=\chunk{\theta}{0}{2p+1}$ with $\theta_{2p+1}=r$, for all $i\neq 2p+1$, $\PPY\as$,
\begin{align*}
\frac{\partial\ell_{z,k}^\theta}{\partial\theta_i}&=\left(\frac{Y_k}{\boldsymbol\delta_p^T\F{\chunk{Y}{1}{k-1}}(z)}-\frac{Y_k+r}{1+\boldsymbol\delta_p^T\F{\chunk{Y}{1}{k-1}}(z)}\right) \boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{Y}{1}{k-1}}(z) \\
\frac{\partial\ell_{z,k}^\theta}{\partial \theta_{2p+1}}&= \Gamma_2(Y_k+r)-\Gamma_2(r)-\ln\left(1+\boldsymbol\delta_p^T\F{\chunk{Y}{1}{k-1}}(z)\right)
\end{align*}
and

\begin{align*}
\frac{\partial\ell_{k}^\theta}{\partial\theta_i}&=\left(\frac{Y_k}{\boldsymbol\delta_p^T\F{\chunk{Y}{-\infty}{k-1}}}-\frac{Y_k+r}{1+\boldsymbol\delta_p^T\F{\chunk{Y}{-\infty}{k-1}}}\right) \boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{Y}{-\infty}{k-1}} \\
\frac{\partial\ell_{k}^\theta}{\partial \theta_{2p+1}}&= \Gamma_2(Y_k+r)-\Gamma_2(r)-\ln\left(1+\boldsymbol\delta_p^T\F{\chunk{Y}{-\infty}{k-1}}\right),
\end{align*}
where $\Gamma_2$ is the digamma function defined by, for all $u>0$,
 \begin{equation*}
\Gamma_2(u)\eqdef\frac{\rmd}{\rmd u}\ln \Gamma (u)=\frac{\int_0^\infty\rme^{-t}t^{u-1}\ln t\,\rmd t}{\int_0^\infty\rme^{-t}t^{u-1}\rmd t}.
\end{equation*}
For $u\ge3$, the digamma function satisfies
 \begin{align}
 \nonumber
\left|\Gamma_2(u)\right|&\le\frac{\int_0^\infty\rme^{-t}t^{u-1}|\ln t|\,\rmd t}{\Gamma(u)}\le\frac{\int_0^\infty\rme^{-t}t^{u-1}(t+t^{-1})\,\rmd t}{\Gamma(u)}\\ \label{eq:def:digamma:gen:od}
&=\frac{\Gamma(u+1)+\Gamma(u-1)}{\Gamma(u)}=u+(u-1)^{-1}\le u+1,
\end{align}
where we have used that $|\ln t|\le (t+t^{-1})$ whenever $t>0$.

It is straight forward to have that, for all $i$, $\PPY\as$,

\begin{align*}
\left|\frac{\partial\ell_{z,k}^\theta}{\partial\theta_i}-\frac{\partial\ell_{k}^\theta}{\partial\theta_i}\right| &\le
\left| \boldsymbol\delta_p^T\F{\chunk{Y}{-\infty}{k-1}}- \boldsymbol\delta_p^T\F{\chunk{Y}{1}{k-1}}(z)\right| \\
&+ \Bigg|\frac{Y_k+r}{1+\boldsymbol\delta_p^T\F{\chunk{Y}{1}{k-1}}(z)}-\frac{Y_k}{\boldsymbol\delta_p^T\F{\chunk{Y}{1}{k-1}}(z)}-\\
&\frac{Y_k+r}{1+\boldsymbol\delta_p^T\F{\chunk{Y}{-\infty}{k-1}}}+\frac{Y_k}{\boldsymbol\delta_p^T\F{\chunk{Y}{-\infty}{k-1}}}\Bigg|\left| \boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{Y}{-\infty}{k-1}}\right| \\
&+\left|\frac{Y_k+r}{1+\boldsymbol\delta_p^T\F{\chunk{Y}{1}{k-1}}(z)}-\frac{Y_k}{\boldsymbol\delta_p^T\F{\chunk{Y}{1}{k-1}}(z)}\right|\\
&\times\left| \boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{Y}{-\infty}{k-1}}- \boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{Y}{1}{k-1}}(z)\right|\eqsp.
\end{align*}
Since $\delta_p^T\F{\chunk{Y}{1}{k-1}}(z)\ge\underline{\omega}$ and so that $\delta_p^T\F{\chunk{Y}{-\infty}{k-1}}\ge\underline{\omega}$, we obtain that for all $i$, $\PPY\as$,

\begin{align*}
&\left|\frac{\partial\ell_{z,k}^\theta}{\partial\theta_i}-\frac{\partial\ell_{k}^\theta}{\partial\theta_i}\right| \le
\left| \boldsymbol\delta_p^T\F{\chunk{Y}{-\infty}{k-1}}- \boldsymbol\delta_p^T\F{\chunk{Y}{1}{k-1}}(z)\right| \\
&+\left(\frac{2Y_k+r}{\underline{\omega}}\right)\left| \boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{Y}{-\infty}{k-1}}\right|\left| \boldsymbol\delta_p^T\F{\chunk{Y}{-\infty}{k-1}}- \boldsymbol\delta_p^T\F{\chunk{Y}{1}{k-1}}(z)\right| \\
&+\left(\frac{2Y_k+r}{\underline{\omega}}\right)\left| \boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{Y}{-\infty}{k-1}}- \boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{Y}{1}{k-1}}(z)\right|\eqsp.
\end{align*}
Note that for all $z\in\Zset$ and $k\ge 1$, we have, $\PPY\as$,
\begin{align*}
\F{\chunk{Y}{-\infty}{k-1}}-\F{\chunk{Y}{1}{k-1}}(z)=\mathbf{A}^{k-1}\left(\F{\chunk{Y}{-\infty}{0}}-z\right)\eqsp,
\end{align*}
$\frac{\partial}{\partial\theta_{2p+1}}\F{\chunk{Y}{1}{k-1}}(z)=\frac{\partial}{\partial\theta_{2p+1}}\F{\chunk{Y}{-\infty}{k-1}}=0$ and, for all $i\neq 2p+1$,
\begin{align*}
\frac{\partial}{\partial\theta_i}\F{\chunk{Y}{-\infty}{k-1}}&-\frac{\partial}{\partial\theta_i}\F{\chunk{Y}{1}{k-1}}(z) \\
&=\frac{\partial(\mathbf{A}^{k-1})}{\partial\theta_i}\left(\F{\chunk{Y}{-\infty}{0}}-z\right)+\mathbf{A}^{k-1}\frac{\partial}{\partial\theta_i}\F{\chunk{Y}{-\infty}{0}}\eqsp.
\end{align*}
By proceeding similarly as in the proof of \autoref{lem:exist:deriv:psi:od} and noting that $\sup_{\theta\in\Theta}\left|\boldsymbol\delta_p^T\F{\chunk{Y}{-\infty}{0}}\right|$ and $\sup_{\theta\in\Theta}\left|\boldsymbol\delta_p^T\frac{\partial}{\partial\theta}\F{\chunk{Y}{-\infty}{0}}\right|$ are finite $\tilde\PP^\thv\as$, then there exist a constant $\rho_1\in(0,1)$ and a $\tilde\PP^\thv\as$ finite random variable $\tilde M_1=\tilde M_1(z)$ such that for all $k\ge 1$, $\tilde\PP^\thv\as$,
\begin{align*}
\left|\boldsymbol\delta_p^T\F{\chunk{Y}{-\infty}{k-1}}-\boldsymbol\delta_p^T\F{\chunk{Y}{1}{k-1}}(z)\right|\le \tilde M_1\rho_1^k\eqsp,
\end{align*}
and for all $i$, $\tilde\PP^\thv\as$,
\begin{align*}
\left|\boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{Y}{-\infty}{k-1}}-\boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{Y}{1}{k-1}}(z)\right|\le \tilde M_1\rho_1^k\eqsp.
\end{align*}

%We also have
%\begin{align}\label{eq:moment:finite:psi:logpois:od:gen}
%\tilde\PE^\thv\left[\sup_{\theta\in\Theta}\left|\F{\chunk{U}{-\infty}{0}}\right|_\infty\right]<\infty
%\end{align}
%and for all $i$
The last two inequalities imply that, for all $\theta\in\Theta$, $i\in\{1,\ldots,2p+2\}$ and $k\in\zsetpnz$, $\PPY\as$,
\begin{align*}
\left|\frac{\partial\ell_{z,k}^\theta}{\partial\theta_i}-\frac{\partial\ell_{k}^\theta}{\partial\theta_i}\right| \le
 \tilde M_1\rho_1^k\left[1+\left(\frac{2Y_k+\bar r}{\underline{\omega}}\right)\left(1+\sup_{\theta\in\Theta} \left| \boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{Y}{-\infty}{k-1}}\right|\right)\right]\eqsp.
\end{align*}
Note also that from \autoref{lem:useful:gen}, we have, for all $i$,
\begin{align}\label{eq:moment:finite:deriv:psi:nbin:od:gen}
\tilde\PE^\thv\left[\sup_{\theta\in\Theta}\left|\boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{Y}{-\infty}{0}}\right|\right]<\infty\eqsp.
\end{align}
Thus, by stationarity and from \eqref{eq:moment:finite:nbin:gen:od} and \eqref{eq:moment:finite:deriv:psi:nbin:od:gen}, \autoref{lem:useful:gen} implies that there exist a constant $\rho_2\in(0,1)$ and a $\tilde\PP^\thv\as$ finite random variable $\tilde M_2=\tilde M_2(z)$ such that for all $\theta\in\Theta$, $i\in\{0,\ldots,2p+1\}$ and $k\ge 1$, $\tilde\PP^\thv\as$,
\begin{align*}
&\left|\frac{\partial\ell_{z,k}^\theta}{\partial\theta_i}-\frac{\partial\ell_{k}^\theta}{\partial\theta_i}\right| \le
\rho_2^k\tilde M_2\eqsp.
\end{align*}
Therefore, Assumption~\ref{item:ctl:limit:od:gen} holds. We now check \ref{item:finte:mement:gen:od}. We have
\begin{align*}
\ell_{1}^\theta&=\ln g^\theta\left(\Proj{p}{\F{\chunk{Y}{-\infty}{0}}} ;Y_1\right)\\
				&= \ln\Gamma(Y_1+r)-(Y_1+r)\ln\left(1+\boldsymbol\delta_p^T\F{\chunk{Y}{-\infty}{0}}\right) \\
			&-Y_1\ln\left(\boldsymbol\delta_p^T\F{\chunk{Y}{-\infty}{0}}\right)-\ln\Gamma(r)-\ln(Y_1!).
\end{align*}
Write $\theta=\chunk{\theta}{0}{2p+1}$. Then for all $i$ and $j$ such that $i\neq 2p+1$ and $j\neq 2p+1$, we have
\begin{align}
\label{eq:ell1:diffi:nbin:gen:od}
\frac{\partial\ell_{1}^\theta}{\partial\theta_i}&=\left(\frac{Y_1}{\boldsymbol\delta_p^T\F{\chunk{Y}{-\infty}{0}}}-\frac{Y_1+r}{1+\boldsymbol\delta_p^T\F{\chunk{Y}{-\infty}{0}}}\right) \boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{Y}{-\infty}{0}}, \\ \label{eq:ell1:diffend:nbin:gen:od}
\frac{\partial\ell_{1}^\theta}{\partial \theta_{2p+2}}&= \Gamma_2(Y_1+r)-\Gamma_2(r)-\ln\left(1+\boldsymbol\delta_p^T\F{\chunk{Y}{-\infty}{0}}\right),
\end{align}
and
\begin{align*}
&\frac{\partial^2\ell_{1}^\theta}{\partial\theta_i\partial\theta_j}= \left(\frac{Y_1}{\boldsymbol\delta_p^T\F{\chunk{Y}{-\infty}{0}}}-\frac{Y_1+r}{1+\boldsymbol\delta_p^T\F{\chunk{Y}{-\infty}{0}}}\right) \boldsymbol\delta_p^T\frac{\partial^2}{\partial\theta_i\partial\theta_j}\F{\chunk{Y}{-\infty}{0}}+\\
&\left(\frac{Y_1+r}{(1+\boldsymbol\delta_p^T\F{\chunk{Y}{-\infty}{0}})^2}-\frac{Y_1}{(\boldsymbol\delta_p^T\F{\chunk{Y}{-\infty}{0}})^2}\right) \boldsymbol\delta_p^T\frac{\partial\F{\chunk{Y}{-\infty}{0}}}{\partial\theta_i}\boldsymbol\delta_p^T\frac{\partial\F{\chunk{Y}{-\infty}{0}}}{\partial\theta_j},\\
&
\frac{\partial^2\ell_{1}^\theta}{\partial\theta_i\partial\theta_{2p+1}}= -\left(1+\boldsymbol\delta_p^T\F{\chunk{Y}{-\infty}{0}}\right)^{-1}\boldsymbol\delta_p^T\frac{\partial\F{\chunk{Y}{-\infty}{0}}}{\partial\theta_i},\\
&\frac{\partial^2\ell_{1}^\theta}{\partial\theta_{2p+1}\partial\theta_{2p+1}}=\Gamma_3(Y_1+r)-\Gamma_3(r),
\end{align*}
where $\Gamma_3$ is the trigamma function defined by, for all $u>0$,
$$
\Gamma_3(u)\eqdef\frac{\rmd}{\rmd u}\Gamma_2(u)=\frac{\int_0^\infty\rme^{-t}t^{u-1}|\ln t|^2\,\rmd t}{\Gamma(u)}-\Gamma_2(u)^2\eqsp.
$$
Note that when $u\ge 3$, the trigamma function satisfies
 \begin{align}
 \nonumber
\left|\Gamma_3(u)\right|&\le\frac{\int_0^\infty\rme^{-t}t^{u-1}(t+t^{-1})^2\,\rmd t}{\Gamma(u)}+\Gamma_2(u)^2\\ \label{eq:def:ineq:gamma3:gen:od}
&=\frac{\Gamma(u+2)+2\Gamma(u)+\Gamma(u-2)}{\Gamma(u)}+\Gamma_2(u)^2\le 2(u+1)^2+3\eqsp.
\end{align}

Using similar lines as the proof of \autoref{lem:exist:deriv:psi:od} together with \eqref{eq:finite:second:mement:Y:nbin:gen:od}, it can be shown that, for all $i$ and $j$,
\begin{align}\label{eq:finite:moment:lim:Psi:gen:od}
\tilde\PE^\thv\left[\sup_{\theta\in\Theta}\left|\frac{\partial}{\partial\theta_i}\F{\chunk{Y}{-\infty}{0}}\right|^2\right]<\infty
\end{align}
and
\begin{align}\label{eq:finite:moment:diff:lim:Psi:gen:od}
\tilde\PE^\thv\left[\sup_{\theta\in\Theta}\left|\frac{\partial^2}{\partial\theta_i\partial\theta_j}\F{\chunk{Y}{-\infty}{0}}\right|\right]<\infty.
\end{align}
From~\eqref{eq:ell1:diffi:nbin:gen:od}, \eqref{eq:ell1:diffend:nbin:gen:od} and  \eqref{eq:def:digamma:gen:od}, we have, for all $i\neq 2p+1$, $\PPY\as$,
\begin{align*}
\left|\frac{\partial\ell_{1}^\theta}{\partial\theta_i}\right|&\le\left(\frac{2Y_k+\bar r}{\underline{\omega}}\right)\sup_{\theta\in\Theta}\left|\boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{Y}{-\infty}{0}}\right|, \\
\left|\frac{\partial\ell_{1}^\theta}{\partial \theta_{2p+2}}\right|&\le Y_1+\tilde C+\ln\left(1+\sup_{\theta\in\Theta}\left|\boldsymbol\delta_p^T\frac{\partial}{\partial\theta_i}\F{\chunk{Y}{-\infty}{0}}\right|\right),
\end{align*}
where $\tilde C$ is a positive constant. Using tower property, \eqref{eq:finite:second:mement:Y:nbin:gen:od} and \eqref{eq:finite:moment:lim:Psi:gen:od}, we obtain that for all $i$,
\begin{align*}
\tilde\PE^\thv\left[\sup_{\theta\in\Theta}\left|\frac{\partial\ell_{1}^\theta}{\partial\theta_i}\right|^2\right]<\infty\eqsp.
\end{align*}
By similar argument, we can obtain from \eqref{eq:finite:second:mement:Y:nbin:gen:od}, \eqref{eq:def:ineq:gamma3:gen:od}, \eqref{eq:finite:moment:lim:Psi:gen:od} and \eqref{eq:finite:moment:diff:lim:Psi:gen:od} that, for all $i$ and $j$,
\begin{align*}
\tilde\PE^\thv\left[\sup_{\theta\in\Theta}\left|\frac{\partial^2\ell_{1}^\theta}{\partial\theta_i\partial\theta_j}\right|\right]<\infty\eqsp.
\end{align*}
Thus, \ref{item:finte:mement:gen:od}-\ref{item:1:finte:moment:od:gen} and \ref{item:finte:mement:gen:od}-\ref{item:2:finte:moment:od:gen} hold. Note that for all $i, j$, we have, $\PPY\as$,
\begin{align*}
\frac{\partial}{\partial\theta_i}g^\theta\left(\Proj{p}{\F{\chunk{Y}{-\infty}{0}}} ;Y_1\right)=g^\theta\left(\Proj{p}{\F{\chunk{Y}{-\infty}{0}}} ;Y_1\right)\frac{\partial\ell_{1}^\theta}{\partial\theta_i}
\end{align*}
and
\begin{align*}
&\frac{\partial^2}{\partial\theta_i\partial\theta_j}g^\theta\left(\Proj{p}{\F{\chunk{Y}{-\infty}{0}}} ;Y_1\right) \\
&=g^\theta\left(\Proj{p}{\F{\chunk{Y}{-\infty}{0}}} ;Y_1\right)\frac{\partial^2\ell_{1}^\theta}{\partial\theta_i\partial\theta_j}+\frac{\partial}{\partial\theta_i}g^\theta\left(\Proj{p}{\F{\chunk{Y}{-\infty}{0}}} ;Y_1\right)\frac{\partial\ell_{1}^\theta}{\partial\theta_j}
\end{align*}
Then, \ref{item:finte:mement:gen:od}-\ref{item:3:finte:moment:od:gen} follows by observing that for all $i$, $\PPY\as$,
\begin{align*}
\left|\frac{\partial}{\partial\theta_i}g^\theta\left(\Proj{p}{\F{\chunk{Y}{-\infty}{0}}} ;Y_1\right)\right|&=g^\theta\left(\Proj{p}{\F{\chunk{Y}{-\infty}{0}}} ;Y_1\right)\left|\frac{\partial\ell_{1}^\theta}{\partial\theta_i}\right|\le\sup_{\theta\in\Theta}\left|\frac{\partial\ell_{1}^\theta}{\partial\theta_i}\right|
\end{align*}
and
\begin{align*}
\tilde\PE^\thv\left[\sup_{\theta\in\Theta}\left|\frac{\partial\ell_{1}^\theta}{\partial\theta_i}\right|\right]\le\left(\tilde\PE^\thv\left[\sup_{\theta\in\Theta}\left|\frac{\partial\ell_{1}^\theta}{\partial\theta_i}\right|^2\right]\right)^{1/2}<\infty\eqsp.
\end{align*}
Similarly, \ref{item:finte:mement:gen:od}-\ref{item:4:finte:moment:od:gen} holds. For concluding, it remains to check \ref{item:inverse:fisher:gen:od}. By \autoref{rem:hessian:invet:gen:od}, it is sufficient to show that \eqref{eq:rem:linear:independent:d-logpois:gen:od} holds. Now let $\boldsymbol{\alpha}=(\alpha_0,\ldots,\alpha_{2p+1})\in\rset^{2p+2}$ and for convenience, write $\theta=(\theta_0,\ldots,\theta_{2p+1})=(\omega,\chunk{a}{1:p},\chunk{b}{2}{p},b_1,r)$. We have $\PPY\as$,

\begin{align*}
\boldsymbol{\alpha}^T\frac{\partial\ell_{1}^\thv}{\partial\theta}&=\left(\frac{Y_1}{\boldsymbol\delta_p^T\F[\thv]{\chunk{Y}{-\infty}{0}}}-\frac{Y_1+r^\star}{1+\boldsymbol\delta_p^T\F[\thv]{\chunk{Y}{-\infty}{0}}}\right)\sum_{i=0}^{2p}\alpha_i \boldsymbol\delta_p^T\frac{\partial\F[\thv]{\chunk{Y}{-\infty}{0}}}{\partial\theta_i} \\
&+\alpha_{2p+1} \left[\Gamma_2(Y_1+r^\star)-\Gamma_2(r^\star)-\ln\left(1+\boldsymbol\delta_p^T\F[\thv]{\chunk{Y}{-\infty}{0}}\right)\right].
\end{align*}

Since $Y_1$ conditionally on $\sigma(\chunk{Y}{-\infty}{0})$ is a negative binomial variable, then $\tilde\PP^\thv\as$, $\boldsymbol{\alpha}^T\frac{\partial\ell_{1}^\thv}{\partial\theta}=0$  implies
$$
\alpha_{2p+1} \left[\Gamma_2(r^\star)+\ln\left(1+\boldsymbol\delta_p^T\F[\thv]{\chunk{Y}{-\infty}{0}}\right)\right]=0,\quad\PPY\as
$$
Since $X_1^\thv=\boldsymbol\delta_p^TZ_1^\thv=\boldsymbol\delta_p^T\F[\thv]{\chunk{Y}{-\infty}{0}}$ is a random variable, then $\alpha_{2p+1}=0$. Thus, $\boldsymbol{\alpha}^T\frac{\partial\ell_{1}^\thv}{\partial\theta}=0$, $\tilde\PP^\thv\as$, also implies
\begin{align*}%\label{eq:lin:indep:1:gen:od:nbin}
\sum_{i=0}^{2p}\alpha_i \boldsymbol\delta_p^T\frac{\partial\F[\thv]{\chunk{Y}{-\infty}{0}}}{\partial\theta_i}=0,\quad\tilde\PP^\thv\as
\end{align*}
The rest of the proof follows by the same argument as in the proof of  \autoref{theo:ergo-convergence:logpois:gen}--\ref{item:thm:logpois:asymp:normal:gen:od} and is thus omitted here. The proof therefore follows.
%Note that we have
%\begin{align}\label{eq:lin:indep:2:gen:od:nbin}
%\frac{\partial\F[\thv]{\chunk{Y}{-\infty}{0}}}{\partial\theta_{2p}}=\sum_{k=1}^{\infty}Y_{-k}\mathbf{A}_\star^k\boldsymbol{\delta}_p+Y_0\boldsymbol{\delta}_p,\quad\tilde\PP^\thv\as
%\end{align}
%Then, since $Y_0$ conditionally on $\sigma(\chunk{Y}{-\infty}{-1})$ is a negative binomial variable, we deduce from \eqref{eq:lin:indep:1:gen:od:nbin} and \eqref{eq:lin:indep:2:gen:od:nbin} that $\alpha_{2p}=0$.
%Following similar argument, we get, for all $k\ge 1$,
%\begin{align*}%\label{eq:invert:A:Astar:zero:gen:od}
%\sum_{i=1}^{2p-1}\alpha_i\boldsymbol{\delta}_p^T\left(\left.\frac{\partial\mathbf{A}^k}{\partial\theta_i}\right|_{\mathbf{A}=\mathbf{A}_\star}\right)\mathbf{b}_\star=0\eqsp.
%\end{align*}
%By \eqref{eq:invert:A:Astar:zero:gen:od:nbin}, it yields $\chunk{\alpha}{1}{2p-1}=0$. Then from \eqref{eq:lin:indep:1:gen:od:nbin}, we finally have
%\begin{align*}
%\alpha_0\boldsymbol\delta_p^T\frac{\partial}{\partial\theta_0}\F[\thv]{\chunk{Y}{-\infty}{0}}=\alpha_0\boldsymbol\delta_p^T\left(\mathbf{I}_{2p-1}-\mathbf{A}_\star\right)^{-1}\boldsymbol\delta_p=0\eqsp,
%\end{align*}
%which implies $\alpha_0=0$ since $\boldsymbol\delta_p^T\left(\mathbf{I}_{2p-1}-\mathbf{A}_\star\right)^{-1}\boldsymbol\delta_p=(1-\sum_{i=1}^pa_i^\star)^{-1}\neq0$.
%
\end{proof}


\section{Empirical Study}\label{sec:empir:study:gen:od}

\subsection{Sharpness of Ergodicity Condition \texorpdfstring{\eqref{eq:def:par:set:logpois:gen}}{} for the Log-linear Poisson GARCH Model}
We assess the sharpness of our general ergodicity condition \eqref{eq:def:par:set:logpois:gen} for the log-linear Poisson GARCH model by simulating several time series with different values of the condition level
$$\nu_L\eqdef\bigvee_{(k_1,\ldots,k_p)\in\{1,2\}^p}\left\|\prod_{j=1}^p\mathbf{A}_{k_j}\right\|_\infty$$
and in the cases where $p=1$ and $2$. Each generated time series holds 2000 observations. \autoref{pic-SimLP:1} presents three log-linear Poisson GARCH time series of order $(1,1)$ with different parameters: Series ODM(1, 1)-(a) corresponds to the model with parameter $(\omega,a_1,b_1)=(-.03, -.7, 1.2)$ and $\nu_L=.7$; Series ODM(1, 1)-(b) corresponds to the model with parameter $(\omega,a_1,b_1)=(-.03, .57, .38)$ and $\nu_L=.95$; and Series ODM(1, 1)-(c) corresponds to the model with parameter $(\omega,a_1,b_1)=(-.03, .603, .402)$ and $\nu_L=1.005$. Remark that in Series ODM(1, 1)-(a), we allow the parameter $b_1$ to be greater than one. \autoref{pic-SimLP:2} exhibits three log-linear Poisson GARCH time series of order $(2,2)$ with different parameters. Series ODM(2, 2)-(a) corresponds to the model with parameter $(\omega,a_1,a_2,b_1,b_2)=(-.02, .4, .1, .3, .1)$ and $\nu_L=.9$; Series ODM(2, 2)-(b) corresponds to the model with parameter $(\omega,a_1,a_2,b_1,b_2)=(-.02, .44, .11, .33, .11)$ and $\nu_L=.99$; and Series ODM(2, 2)-(c) corresponds to the model with parameter $(\omega,a_1,a_2,b_1,b_2)=(-.02, .446, .1115, .3345, .1115)$ and $\nu_L=1.0062$. For both \autoref{pic-SimLP:1} and \autoref{pic-SimLP:2}, it turns out that when the condition level $\nu_L$ is far away from its boundary ($\nu_L<1$), the processes appear quite stable. However, the level of instability seems to increase as the condition level $\nu_L$ gets closer to one. When this boundary is passed, the models appear to be unstable. We should remark that when $\nu_L<1$, the stable behavior of these processes is persistent regardless of their starting points.


\begin{figure}[H]
\centering
\includegraphics[scale=0.8,center]{SimLP1.pdf}
\caption[Simulated Time Series for Log-linear Poisson GARCH of Order $(1,1)$]{Simulated time series for the log-linear Poisson GARCH of order $(1,1)$. Series ODM(1,1)-(a), -(b) and -(c) corresponds to the parameters $(\omega,a_1,b_1)$ equal to $(-.03, -.7, 1.2)$, $(-.03, .57, .38)$ and $(-.03, .603, .402)$, respectively.}\label{pic-SimLP:1}
\end{figure}

\begin{figure}[H]
\centering
%\textbf{The behavior of the \gls{mle}}
\includegraphics[scale=0.8,center]{SimLP2.pdf}
\caption[Simulated Time Series for Log-linear Poisson GARCH of Order $(2,2)$]{Simulated time series for the log-linear Poisson GARCH of order $(2,2)$. Series ODM(2,2)-(a), -(b) and -(c) corresponds to the parameters $(\omega,a_1,a_2,b_1,b_2)$ equal to $(-.02, .4, .1, .3, .1)$, $(-.02, .44, .11, .33, .11)$ and $(-.02, .446, .1115, .3345, .1115)$, respectively.}\label{pic-SimLP:2}
\end{figure}


\subsection{Data Example}
Our purpose here is to give some empirical evidence on some particular time series data for which higher-order \acrshortpl{odm} may be more appropriate than the first-order ones. Our data set, provided by U.S. Geology Survey (USGS) (downloadable on: \url{http://earthquake.usgs.gov/earthquakes/search/}), contains a time series of numbers of earthquakes occurring hourly around the globe with magnitudes larger than 4.0 Richters (a level that is detectable by humans), consisting of 8759 observations during the year 2014, from January 1 to December 31.
%The values of the sample average and the sample variance of this time series are $2.003$ and $3.249$, respectively, suggesting that the data is over dispersed.
The time series and its serial correlation plots are shown in \autoref{pic-earthquake:series} and \autoref{pic-acf:series}, respectively, exhibiting that data possesses a long memory. In our analysis, we used the first 8559 observations for conducting the parameter estimation and the last 200 observations for evaluating the prediction performance of the fitted models. We fitted the data by several different models, by the \gls{nbingarch} and the log-linear Poisson GARCH models with different orders: $(1,1), (1,2), (2,1), (2,2), (3,1), (3,2)$ and $(3,3)$. For estimating the parameters, we use the constrained nonlinear optimization function \texttt{auglag}
(Augmented Lagrangian Minimization Algorithm) from the package \texttt{alabama}
(Augmented Lagrangian Adaptive Barrier Minimization Algorithm) in \texttt{R}. The values of AIC, BIC and average squared prediction error (ASPE), $\text{ASPE}=m^{-1}\sum_{k=1}^m(Y_{k}-\hat Y_{k})^2$, where $\hat Y_k$ is the predictor of the observation $Y_k$---hence the mean process evaluated at the \acrshort{mle}, for these models are reported in ~\autoref{table:aic:bic:equake}. It shows that, using either the AIC or BIC criterion, the higher-order models are preferred for both the \gls{nbingarch} and the log-linear Poisson GARCH, opposed to their first-order counterparts. Among the fitted models, for the \gls{nbingarch}, the AIC suggests the model of order $(3,3)$ corresponding to the estimator $(\hat\omega,\hat a_1,\hat a_2,\hat a_3,\hat b_1,\hat b_2,\hat b_3, \hat r)=(.035, .424, .068, .222, .021,1.036\times10^{-8},5.681\times10^{-8},7.431)$, while the BIC  favors the model of order $(3,1)$ corresponding to the estimated parameter $(\hat\omega,\hat a_1,\hat a_2,\hat a_3,\hat b_1, \hat r)=(.036, .414, .067, .232, .022, 7.131)$. We observe that the values of $\hat b_2$ and $\hat b_3$ are so small that they might be insignificant to be included. However, to discard these coefficients from the model properly, we may need to conduct a hypothesis testing and this is beyond the scope of this paper. Meanwhile, let us keep these terms in the model for convenience. For the log-linear Poisson GARCH models, the AIC suggests the model of $(3,3)$ corresponding to the estimator $(\hat\omega,\hat a_1,\hat a_2,\hat a_3,\hat b_1,\hat b_2,\hat b_3)=(-.043, .495, .142, .091, .256,1.084\times10^{-5}, -.017)$, whereas the BIC picks the model with order $(2,1)$ corresponding to the estimator $(\hat\omega,\hat a_1,\hat a_2,\hat b_1)=(-.041, .471, .234, .251)$. The estimators for the \gls{nbingarch}$(1,1)$ and for the log-linear Poisson GARCH$(1,1)$ are $(\hat\omega,\hat a_1,\hat b_1,\hat r)=(.041, .719, .019, 6.997)$ and $(\hat\omega,\hat a_1,\hat b_1)=(-.028, .722, .226)$, respectively. ~\autoref{table:aic:bic:equake} also shows that the ASPE values of the models selected by either the AIC or the BIC criterion are lower than those of the first-order ones. This suggests that the higher-order models outperform in both cases of the \gls{nbingarch} and the log-linear Poisson GARCH. However, there are barely any differences between these values for the models selected by the AIC and  by the BIC, respectively. By comparing across the models, the selected log-linear Poisson GARCH model seems to yield prediction performance as well as the selected \gls{nbingarch} model  in our data. \autoref{pic-nbingarch:series} and \autoref{pic-logpois:series} respectively depict the predicted series of first-order models and the ones chosen by AIC and BIC, compared with the real observations, for the \gls{nbingarch} and the log-linear Poisson GARCH.

\begin{table}[h]
\caption[AIC, BIC and ASPE Values for the NBIN-GARCH and the Log-linear Poisson GARCH with Different Orders]{AIC, BIC and ASPE values for the \gls{nbingarch} and the log-linear Poisson GARCH models with different orders: $(1,1)$, $(1,2)$, $(2,1)$, $(2,2)$, $(3,1)$, $(3,2)$ and $(3,3)$.}\label{table:aic:bic:equake}
\begin{center}
\smallskip\noindent
\resizebox{\linewidth}{!}{
\begin{tabular}{|r | r r r r r r r l}
\cline{1-8}					
 	       & \multicolumn{7}{c}{NBIN-GARCH} \\ \cline{2-8}
               & $(1,1)$ & $(1,2)$  & $(2,1)$ & $(2,2)$  & $(3,1)$ & $(3,2)$ & $(3,3)$   \\ \cline{1-8}
 \multicolumn{1}{|c|}{AIC} & 30102  &  30097 & 30083 & 30078 & 30074 & 30070 & \textbf{30065}   \\ \cline{1-8}
 \multicolumn{1}{|c|}{BIC} & 30130  &  30132 & 30118 & 30120 & \textbf{30117} & 30119 & 30122   \\ \cline{1-8}
  \multicolumn{1}{|c|}{ASPE} &1.633 &1.633 &1.618 &1.633 &1.614 &1.614 &1.613   \\ \cline{1-8}
               & \multicolumn{7}{|c}{Log-linear Poisson GARCH} \\ \cline{2-8}
               & $(1,1)$ & $(1,2)$  & $(2,1)$ & $(2,2)$ &  $(3,1)$ & $(3,2)$ & $(3,3)$   \\ \cline{1-8}
\multicolumn{1}{|c|}{AIC} & 30632  & 30621 &  30616 & 30614 & 30610 & 30608 & \textbf{30607}   \\ \cline{1-8}
\multicolumn{1}{|c|}{BIC} & 30653  & 30649 & \textbf{30644.5}  & 30649 & 30645.4 & 30651 & 30656   \\ \cline{1-8}
\multicolumn{1}{|c|}{ASPE} &1.654 &1.658 &1.616 &1.616 &1.613 &1.615 &1.617   \\ \cline{1-8}

\end{tabular}
}
\end{center}
\end{table}

%\begin{table}[h]
%\caption[Comparison of Average Squared Prediction Error (ASPE)]{Comparison of average squared prediction errors (ASPEs) for the \gls{nbingarch}$(1,1)$, the log-linear Poisson GARCH$(1,1)$ and the corresponding ones selected by AIC.\label{table:mse:equake}}
%\begin{center}
%\smallskip\noindent
%\resizebox{\linewidth}{!}{
%\begin{tabular}{|r | r r r r r r r l}
%\cline{1-8}					
% 	       & \multicolumn{7}{c}{NBIN-GARCH} \\ \cline{2-8}
%  & $(1,1)$ & $(1,2)$  & $(2,1)$ & $(2,2)$  & $(3,1)$ & $(3,2)$ & $(3,3)$   \\ \cline{1-8}
% \multicolumn{1}{|c|}{APSE} &1.633 &1.633 &1.618 &1.633 &1.614 &1.614 &1.613   \\ \cline{1-8}
%               & \multicolumn{7}{|c}{Log-linear Poisson GARCH} \\ \cline{2-8}
%  & $(1,1)$ & $(1,2)$  & $(2,1)$ & $(2,2)$ &  $(3,1)$ & $(3,2)$ & $(3,3)$   \\ \cline{1-8}
%\multicolumn{1}{|c|}{APSE} &1.654 &1.658 &1.616 &1.616 &1.613 &1.615 &1.617   \\ \cline{1-8}
%
%\end{tabular}
%}
%\end{center}
%\end{table}

\begin{figure}[H]
\centering
%\textbf{The behavior of the \gls{mle}}
\includegraphics[scale=0.6,center]{EarthquakeSeriesFull.pdf}
\caption[Time Series of Number of Earthquakes in 2014 (Hourly)]{Time series of numbers of earthquakes in 2014, from January 1 to December 31 (hourly).}\label{pic-earthquake:series}
\end{figure}

\begin{figure}[H]
\centering
%\textbf{The behavior of the \gls{mle}}
\includegraphics[scale=0.6,center]{acfEarthquake.pdf}
\caption[Serial Autocorrelation for Earthquake Data]{Sample autocorrelation function.}\label{pic-acf:series}
\end{figure}

\begin{figure}[H]
\centering
%\textbf{The behavior of the \gls{mle}}
\includegraphics[scale=0.7,height=7.5cm,center]{nbingarch_p_q.pdf}
\caption[Prediction of NBIN-GARCH models]{Predicted series by the NBIN-GARCH of order $(1,1)$ (dashed blue), by the one selected by the AIC (long dashed red) and by the one selected by the BIC (small dotted green), compared to the true series (black).}\label{pic-nbingarch:series}
\end{figure}

\begin{figure}[H]
\centering
%\textbf{The behavior of the \gls{mle}}
\includegraphics[scale=0.7,height=7.5cm,center]{logpois_p_q.pdf}
\caption[Prediction of Log-linear Poisson GARCH Models]{Predicted series by the log-linear Poisson GARCH of order $(1,1)$ (dashed blue), by the one selected by the AIC (long dashed red) and by the one selected by the BIC (small dotted green), compared to the true series (black).}\label{pic-logpois:series}
\end{figure}


\section{Postponed Proofs}\label{sec:append:proofs:gen:od}
%\subsection{Convergence of the MLE}
%\label{sec:convergence-mle}
%
%Assumptions~\ref{assum:gen:identif:unique:pi:gen} and~\ref{ass:21-lyapunov:gen:od} are
%supposed to hold throughout this section.
%The proof of \autoref{thm:convergence-main:gen:od} relies on the approach introduced
%in \cite{pfanzagl:1969}, which was already used in \cite{dou:kou:mou:2013} for
%a restricted class of observation-driven models.
%Our main contribution here is
%to provide the handy conditions listed in
%Assumption~\ref{assum:practical-cond:O:gen:od}. We first show that our conditions
%imply~\ref{assum:Ptheta:thetastar:is:density:gen:od} and the following one.
%\begin{hyp}{B}
%\item \label{assum:limit:Y} There exists $x_1\in\Xset$ such that, for all
%  $\theta,\thv\in\Theta$, $p^{\theta}(Y_1\,|\,\chunk{Y}{-\infty}{0})$
%  defined as in~(\ref{eq:def-p-theta-neq-thv:gen:od}) with $x=x_1$ is finite $\tilde\PP^{\thv}\as$ Moreover, for all $\thv\in\Theta$, we
%  have
%\begin{align}\label{eq:unfi-limit-contrast-od}
%\lim_{k \to \infty} \sup_{\theta \in \Theta} \left|\ln \frac{g^\theta (\f{\chunk{Y}{1}{k-1}}(x_1);Y_k)}{p^{\theta}(Y_k\,|\,\chunk{Y}{-\infty}{k-1})}\right| =0\quad\tilde\PP^{\thv}\as
%\end{align}
%\end{hyp}
%Indeed we have the following lemma.
%\begin{lemma}\label{lem:exist:practical-cond:O}
%Assumptions \ref{assum:continuity:Y:g-theta:gen:od}, \ref{assum:continuity:Y:phi-theta:gen:od} and~\ref{assum:practical-cond:O:gen:od}
%imply~\ref{assum:limit:Y} and~\ref{assum:Ptheta:thetastar:is:density:gen:od}.
%\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{proof}
%See \ref{app:lem}.
%\end{proof}
%Now the proof of \autoref{thm:convergence-main:gen:od} directly follows from the following
%lemma.
%\begin{lemma}\label{lemma:gen:convergence-od}
%  Assume that
%  \ref{assum:continuity:Y:g-theta:gen:od},
%  \ref{assum:continuity:Y:phi-theta:gen:od} and  \ref{assum:practical-cond:O:gen:od}\ref{assum:exist:practical-cond:subX:gen:od}--\ref{assum:momentCond:gen:od} hold and that $x_1$
%  satisfies~\ref{assum:limit:Y}.  Then $\Theta_\star$ defined
%  by~(\ref{eq:def-Theta-star-set:gen:od}) is a non-empty closed subset of $\Theta$ and
%  \eqref{eq:strong-consistency-prelim:gen:od} holds.
%\end{lemma}
%\begin{proof}  By \cite[Theorem~33]{dou:kou:mou:2013}, to obtain~(\ref{eq:strong-consistency-prelim:gen:od}), it is sufficient
%  to show that, for all $\thv\in\Theta$, the two following assertions hold.
%\begin{enumerate}[label=(\alph*)]
%\item\label{item:finite:moment}
%$\tilde\PE^{\thv} \left[\sup_{\theta\in\Theta}\lnp p^{\theta}(Y_1\,|\,\chunk{Y}{-\infty}{0})\right]<\infty$\,,
%\item\label{item:con:limit}
%the function $\theta\mapsto\ln p^{\theta}(Y_1\,|\,\chunk{Y}{-\infty}{0})$ is continuous on
%$\Theta$, $\tilde\PP^{\thv}\as$
%\end{enumerate}
%In~\ref{assum:limit:Y}, $p^{\theta}(Y_1\,|\,\chunk{Y}{-\infty}{0})$ is
%defined $\tilde\PP^{\thv}\as$ as the limit in~(\ref{eq:def-p-theta-neq-thv:gen:od}) with $x=x_1$. So, $\tilde\PP^{\thv}\as$,
%by~\ref{assum:practical-cond:O:gen:od}\ref{assum:exist:practical-cond:subX:gen:od}--\ref{assum:momentCond:gen:od},
%$p^{\theta}(Y_1\,|\,\chunk{Y}{-\infty}{0})$ is bounded by
%the finite constant appearing in~\ref{assum:practical-cond:O:gen:od}\ref{assum:momentCond:gen:od}.
%Hence Condition~\ref{item:finite:moment} holds.
%
%Condition \ref{item:con:limit} then follows from~(\ref{eq:unfi-limit-contrast-od}).
%Since almost sure convergence implies the convergence in probability and $\tilde\PP^{\thv}$ is shift invariant,
% the random sequence
% \begin{align*}
%U_p\eqdef\sup_{\theta \in \Theta} \left|\ln \frac{g^{\theta} (\f{\chunk{Y}{-m}{0}}(x_1);Y_1)}{p^{\theta}(Y_1\,|\,\chunk{Y}{-\infty}{0})}\right|\;,\quad{m\in\zsetp}\;,
%\end{align*}
%converges to zero in $\tilde\PP^{\thv}$-probability. Then there exists a
%subsequence of $(U_p)$ which converges $\tilde\PP^{\thv}\as$ to zero. Hence,
%interpreting this convergence as a uniform (in $\theta$) convergence of $\ln
%g^{\theta} (\f{\chunk{Y}{-m}{0}}(x_1);Y_1)$ to $\ln
%{p^{\theta}(Y_1\,|\,\chunk{Y}{-\infty}{0})}$ to conclude
%that~\ref{item:con:limit} holds, it is sufficient to show that
%$\theta\mapsto\ln g^{\theta} (\f{\chunk{Y}{-m}{0}}(x_1);Y_1)$ is continuous for
%all $m$ $\tilde\PP^{\thv}\as$ This is indeed the case by~\ref{assum:continuity:Y:g-theta:gen:od}
%and~\ref{assum:continuity:Y:phi-theta:gen:od} and since $g^\theta(x;y)$
%is positive.
%\end{proof}
%% , the function $\theta\mapsto\ln{g^\theta
%%   (\F[\theta]{\chunk{y}{-m}{0}}(x);Y_1)}$ is continuous for all $m\in\zsetp$
%% and $\chunk{y}{-m}{1}\in\Yset^m$.
%
%
%% To obtain \ref{item:con:limit}, it is worth noting that
%% Assumption~\ref{assum:exist:x0:O} rooted in~\ref{assum:limit:Y} implies
%% with~\ref{assum:continuity:Y:g-theta:gen:od} that for all $\theta, \thv\in\Theta$,
%% $\tilde\PP^{\thv}\as$,
%%   \begin{align}\label{eq:log-g-lilmite}
%% \lim_{m\to\infty}\ln g^\theta(\F[\theta]{\chunk{Y}{-m}{0}}(x); Y_1)=\ln g^\theta(\F[\theta,\thv]{\chunk{Y}{-\infty}{0}}; Y_1)\eqsp.
%% \end{align}
%% Since almost sure convergence implies the convergence in probability, since
%%  $\tilde\PP^{\thv}$ is shift invariant, \ref{assum:limit:Y} implies that the random sequence
%%  \begin{align*}
%% U_p\eqdef\sup_{\theta \in \Theta} \left|\ln \frac{p^{\theta,\thv} (\f{\chunk{Y}{-m}{0}}(x);Y_1)}{g^\theta (\F[\theta,\thv]{\chunk{Y}{-\infty}{0}};Y_1)}\right|\;,\quad{m\in\zsetp}\;,
%% \end{align*}
%% converges to zero in $\tilde\PP^{\thv}$-probability. Then there exists a
%% subsequence of $(U_p)$ which converges
%% $\tilde\PP^{\thv}\as$ to zero. Hence,
%% since from~\ref{assum:continuity:Y:g-theta:gen:od} and~\ref{assum:continuity:Y:phi-theta:gen:od}, the function $\theta\mapsto\ln{g^\theta
%%   (\F[\theta]{\chunk{Y}{-m}{0}}(x);Y_1)}$ is continuous
%% for all $m\in\zsetp$, $\thv\in\Theta$ and $\chunk{Y}{-m}{1}$, we conclude
%% with~(\ref{eq:log-g-lilmite}) that~\ref{item:con:limit} holds.
%
%
%%\begin{hyp}{P}
%%\item \label{assum:practical-cond:1} Assumption~\ref{assum:practical-cond:O:gen:od}
%%  hold and
%%\begin{enumerate}[label=(\roman*)]
%%\item \label{item:bar:phi}For all $\theta\in\Theta$, $\pi^\theta_2\left(\lnp (\bar{\phi})\right)<\infty$.
%%\item \label{item:bar:phi:strong}  If $C>0$ then for all $\theta\in\Theta$, $\pi^\theta_2\left(\bar{\phi}\right)<\infty$.
%%\end{enumerate}
%%\end{hyp}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%We will also make use of
%\cite[Lemma~34]{dou:kou:mou:2013} which we quote here for convenience.
%\begin{lemma} \label{lem:useful:gen:od}
%Let \sequence{U}[k][\zsetp] be a stationary sequence of real-valued random variables on $(\Omega, \mcf, \PP)$. Assume that $\PE{[\lnp |U_0|]}<\infty$. Then, for all $\eta \in (0,1)$,
%$$
%\lim_{k \to \infty} \eta^k U_k =0\, , \quad \PP\as
%$$
%\end{lemma}






%\begin{lemma}
%Suppose each of the following holds:
%\begin{enumerate}
%\item
%$\sup_{x,y,\theta}|\lambda|_{\max}\left(\frac{\partial\psi_y^\theta}{\partial x}(x)\right)<1$
%\item For all $x,x',y, \theta$,
%$\left\|\frac{\partial\psi_y^\theta(x)}{\partial\theta}-\frac{\partial\psi_y^\theta(x')}{\partial\theta}\right\|\le M(y)|x-x'|$
%\item For all $x,x',y, \theta$,
%$\left\|\frac{\partial\psi_y^\theta}{\partial x}(x)-\frac{\partial\psi_y^\theta}{\partial x}(x')\right\|\le M(y)|x-x'|$
%
%\item
%$|\f[\theta]{\chunk{y}{1}{n}}(x)-\f[\theta]{\chunk{y}{1}{n}}(x')|\le\varrho_0^n|x-x'|$,
%
%\item
%$\left\|\frac{\partial\f[\theta]{\chunk{y}{1}{n}}(x)}{\partial\theta}-\frac{\partial\f[\theta]{\chunk{y}{1}{n}}(x')}{\partial\theta}\right\|\le\varrho_1^n|x-x'|$,
%
%\item
%$\left\|\frac{\partial^2\f[\theta]{\chunk{y}{1}{n}}(x)}{\partial\theta\partial\theta^T}-\frac{\partial^2\f[\theta]{\chunk{y}{1}{n}}(x')}{\partial\theta\partial\theta^T}\right\|\le\varrho_2^n|x-x'|$,
%
%\end{enumerate}
%then there exists $\f[\theta]{\chunk{y}{-\infty}{0}}$ such that for all $x$,
%\begin{equation*}
%\lim_{m\to\infty}\sup_{\theta}\left\|\frac{\partial\f[\theta]{\chunk{Y}{-m}{0}}(x)}{\partial \theta}-\dot{\psi}^\theta\langle{\chunk{Y}{-\infty}{0}}\rangle\right\|=0\quad\tilde\PP^\thv\as\eqsp,
%\end{equation*}
%and, moreover,
%\begin{equation}
%\frac{\partial\f[\theta]{\chunk{y}{-\infty}{0}}}{\partial \theta}=\dot{\psi}^\theta\langle{\chunk{y}{-\infty}{0}}\rangle\eqsp.
%\end{equation}
%\end{lemma}
%
%\begin{proof}
%We have for all $x$,
%\begin{align*}
%&\frac{\partial\f[\theta]{\chunk{Y}{-m-1}{0}}(x)}{\partial\theta}-\frac{\partial\f[\theta]{\chunk{Y}{-m}{0}}(x)}{\partial\theta} \\
%&=\frac{\partial}{\partial\theta}\left[\f[\theta]{\chunk{Y}{-m}{0}}(\psi_{Y_{-m-1}}^\theta(x))\right]-\frac{\partial\f[\theta]{\chunk{Y}{-m}{0}}(x)}{\partial\theta}\\
%&=\frac{\partial\f[\theta]{\chunk{Y}{-m}{0}}}{\partial\theta}(\psi_{Y_{-m-1}}^\theta(x))+\frac{\partial\f[\theta]{\chunk{Y}{-m}{0}}}{\partial x}(\psi_{Y_{-m-1}}^\theta(x))\frac{\partial \psi_{Y_{-m-1}}^\theta(x)}{\partial\theta}\\
%&-\frac{\partial\f[\theta]{\chunk{Y}{-m}{0}}(x)}{\partial\theta} \\
%&=A_p+B_p,
%\end{align*}
%where
%\begin{align*}
%A_p=\frac{\partial\f[\theta]{\chunk{Y}{-m}{0}}}{\partial\theta}(\psi_{Y_{-m-1}}^\theta(x))-\frac{\partial\f[\theta]{\chunk{Y}{-m}{0}}(x)}{\partial\theta}
%\end{align*}
%and
%\begin{align*}
%B_p&=\frac{\partial\f[\theta]{\chunk{Y}{-m}{0}}}{\partial x}(\psi_{Y_{-m-1}}^\theta(x))\frac{\partial \psi_{Y_{-m-1}}^\theta(x)}{\partial\theta}\\
%&=\frac{\partial\psi_{Y_0}^\theta}{\partial x}(\f[\theta]{\chunk{Y}{-m-1}{-1}}(x))\cdots\frac{\partial\psi_{Y_{-m}}^\theta}{\partial x}(\psi_{Y_{-m-1}}^\theta(x))\frac{\partial \psi_{Y_{-m-1}}^\theta(x)}{\partial\theta}.
%\end{align*}
%From ..., and we have
%\begin{align*}
%\|A_p\|\le\varrho_1^{m+1}\left(\sup_{\theta}\left|\psi_{Y_{-m-1}}^\theta(x)\right|+|x|\right).
%\end{align*}
%and
%\begin{align*}
%\|B_p\|\le\bar\varrho^{m}\left(\sup_{\theta}\left|\frac{\partial \psi_{Y_{-m-1}}^\theta(x)}{\partial\theta}\right|\right),
%\end{align*}
%respectively. Since $\left(\sup_{\theta}\left|\psi_{Y_{-m-1}}^\theta(x)\right|\right)_{m\in\zsetp}$ and $\left(\sup_{\theta}\left|\frac{\partial \psi_{Y_{-m-1}}^\theta(x)}{\partial\theta}\right|\right)_{m\in\zsetp}$ are stationary, they are bounded in $\tilde\PP^\thv$-probability. Then for any $\rho\in(0,1)$, $\lim_{m\to\infty}\rho^m\sup_{\theta}\left|\psi_{Y_{-m-1}}^\theta(x)\right|=0$ and $\lim_{m\to\infty}\rho^m\sup_{\theta}\left|\frac{\partial \psi_{Y_{-m-1}}^\theta(x)}{\partial\theta}\right|=0$ in $\tilde\PP^\thv$-probability. This implies, there exists $\varrho\in(0,1)$ such that $\lim_{m\to\infty}\varrho^{-m}(\|A_p\|+\|B_p\|)=0$ in $\tilde\PP^\thv$-probability. Consequently, $\left(\frac{\partial\f[\theta]{\chunk{Y}{-m}{0}}(x)}{\partial\theta}\right)_{m\in\zsetp}$ is a Cauchy sequence in $\tilde\PP^\thv$-probability and thus a (uniformly) convergent sequence in $\tilde\PP^\thv$-probability.
%\end{proof}

\subsection{Proof of~\autoref{lem:exist:deriv:psi:od}}\label{sec:proof:lem:deriv:psi:od:gen}
\begin{proof}
From~\eqref{eq:def:psi:linear:od}, for all $n\in\zsetpnz,\theta\in\Theta,z\in\Zset$, and $\chunk{y}{1}{n}\in\Yset^n$, we have
\begin{equation}
\F[\theta]{\chunk{y}{1}{n}}(z)=\left(\prod_{\ell=1}^{n}a(\theta,y_{n-\ell+1})\right)z+\sum_{k=1}^n{\left(\prod_{\ell=1}^{k-1}a(\theta,y_{n-\ell})\right)b(\theta,y_{n-k+1})}.
\end{equation}
Then, for all $m\in\zsetp$, $\theta\in\Theta$ and $z\in\Zset$, we get that
$$
\F[\theta]{\chunk{Y}{-m}{0}}(z)=\left(\prod_{\ell=0}^{m}a(\theta,Y_{-\ell})\right)z+\sum_{k=0}^m\left(\prod_{\ell=0}^{k-1}a(\theta,Y_{-\ell})\right)b(\theta,Y_{-k}).
$$
By \ref{assum:phi:linear:od}\ref{item:ineq:prod:rho:n:od}, we have
$$
\lim_{m\to\infty}\sup_{\theta\in\Theta}\left\|\left(\prod_{\ell=0}^{m}a(\theta,Y_{-\ell})\right)z\right\|=0,\quad \PPY\as
$$
Moreover, we have

$$
\sum_{k=0}^\infty\left\|\left(\prod_{\ell=0}^{k-1}a(\theta,Y_{-\ell})\right)b(\theta,Y_{-k})\right\|\le \bar C \sum_{k=0}^\infty\rho^k\sup_{\theta\in\Theta}\left\|b(\theta,Y_{-k})\right\|\eqsp,
$$
for some positive constant $\bar C$. By \ref{assum:phi:linear:od}\ref{item:finite:sup:b:od} and \autoref{lem:useful:gen}, the right-hand side of the the above inequality converges $\PPY\as$ This implies that for all $z\in\Zset$, the sequence $\nsequence{\F[\theta]{\chunk{Y}{-m}{0}}(z)}{m\in\zsetp}$ converges uniformly on $\Theta$ to a limit $\F[\theta]{\chunk{Y}{-\infty}{0}}\eqdef\sum_{k=0}^\infty\left(\prod_{\ell=0}^{k-1}a(\theta,Y_{-\ell})\right)b(\theta,Y_{-k})$, $\PPY\as$, showing that \eqref{eq:def:psi:infty:gen:od} is satisfied. Next we show that  \eqref{eq:def:Z1:Psi:gen:od} is satisfied. Note that for all $m\in\zsetp$
\begin{align*}
\F[\thv]{\chunk{Y}{-m}{0}}(Z_{-m})&=\left(\prod_{\ell=0}^{m}a(\thv,Y_{-\ell})\right)Z_{-m}+\sum_{k=0}^m\left(\prod_{\ell=0}^{k-1}a(\thv,Y_{-\ell})\right)b(\thv,Y_{-k}).
\end{align*}
Since $\nsequence{Z_{-m}}{m\in\zsetp}$ is stationary under $\PP^\thv$ so that it is bounded in probability, and since $\lim_{m\to\infty}\prod_{\ell=0}^{m}a(\thv,Y_{-\ell})=0$, $\PP^\thv\as$, it then implies that $\lim_{m\to\infty}\left(\prod_{\ell=0}^{m}a(\thv,Y_{-\ell})\right)Z_{-m}=0$ in $\PP^\thv$-probability. From the proof above, we have
$$
\lim_{m\to\infty}\sum_{k=0}^m\left(\prod_{\ell=0}^{k-1}a(\thv,Y_{-\ell})\right)b(\thv,Y_{-k})=\F[\thv]{\chunk{Y}{-\infty}{0}},\quad\PP^\thv\as
$$
Note also that for all $m\in\zsetp$, $Z_1=\F[\thv]{\chunk{Y}{-m}{0}}(Z_{-m})$, $\PP^\thv\as$ We therefore get that
\begin{equation*}
\F[\thv]{\chunk{Y}{-\infty}{0}}= Z_1, \quad  \PP^{\thv}\as,
\end{equation*}
showing \eqref{eq:def:Z1:Psi:gen:od}. In addition, \ref{assum:phi:linear:od}\ref{item:finite:sup:b:od} implies
\begin{align*}
\PEY\left[\sup_{\theta\in\Theta}\left\|\F[\theta]{\chunk{Y}{-\infty}{0}}\right\|\right]&\le \bar C \sum_{k=0}^\infty\rho^k\PEY\left[\sup_{\theta\in\Theta}\left\|b(\theta,Y_{-k})\right\|\right]\\
&\le\frac{\bar C}{1-\rho}\PEY\left[\sup_{\theta\in\Theta}\left\|b(\theta,Y_{1})\right\|\right]<\infty\eqsp,
\end{align*}
showing \eqref{eq:finite:F1:gen:od}.

For any $i$, we have

\begin{align*}
&\frac{\partial\F[\theta]{\chunk{Y}{-m}{0}}(z)}{\partial\theta_i} \\
&=\frac{\partial\left[\left(\prod_{\ell=0}^m a(\theta,Y_{-\ell})\right)z\right]}{\partial\theta_i}+\sum_{k=0}^m\frac{\partial\left[\left(\prod_{\ell=0}^{k-1}a(\theta,Y_{-\ell})\right)b(\theta,Y_{-k})\right]}{\partial\theta_i}\\
&=\sum_{\ell'=0}^m\left(\prod_{\ell=0}^{m}a_{\ell', i}(\theta,Y_{-\ell})\right)z + \sum_{k=0}^m \left(\sum_{\ell'=0}^{k-1}\prod_{\ell=0}^{k-1}a_{\ell', i}(\theta,Y_{-\ell})\right)b(\theta,Y_{-k}) \\
&+\sum_{k=0}^m\left(\prod_{\ell=0}^{k-1}a(\theta,Y_{-\ell})\right)\frac{\partial b(\theta,Y_{-k})}{\partial\theta_i},
\end{align*}
where for all $i, \ell, \ell'$,
\begin{equation}\label{eq:def:a:ell:i}
a_{\ell', i}(\theta,Y_{-\ell})=\begin{cases}
 a(\theta,Y_{-\ell}) & \mbox{if}\quad\ell\neq\ell' \\
\frac{\partial}{\partial\theta_i}a(\theta,Y_{-\ell}) & \mbox{if}\quad\ell=\ell'.
\end{cases}
\end{equation}
 From \ref{assum:phi:linear:od}\ref{item:ineq:prod:rho:n:od}, we have
\begin{align*}
&\left\|\sum_{\ell'=0}^m\left(\prod_{\ell=0}^{m}a_{\ell', i}(\theta,Y_{-\ell})\right)z\right\|\le\sum_{\ell'=0}^m\left\|\prod_{\ell=0}^{m}a_{\ell', i}(\theta,Y_{-\ell})\right\|\|z\| \\
&\le\sum_{\ell'=0}^mC^2\rho^{m}\left\|\frac{\partial a(\theta,Y_{-\ell'})}{\partial\theta_i}\right\|\|z\| \\
&\le C^2\|z\|\left(\frac{1+\rho}{2}\right)^m\sum_{\ell'=0}^m\left(\frac{2\rho}{1+\rho}\right)^{\ell'}\sup_{\theta\in\Theta}\left\|\frac{\partial a(\theta,Y_{-\ell'})}{\partial\theta_i}\right\|.
\end{align*}
%Since, from (\ref{item:ineq:first:diff-a-b:finite:od}), $\tilde\PE^\thv\left[\lnp\sup_{\theta\in\Theta}\left\|\frac{\partial a(\theta,Y_{1})}{\partial\theta_i}\right\|\right]<\infty$,
We have from \ref{assum:phi:linear:od}\ref{item:ineq:first:diff-a-b:finite:od} and by \autoref{lem:useful:gen},
$$\sum_{\ell'=0}^\infty\left(\frac{2\rho}{1+\rho}\right)^{\ell'}\sup_{\theta\in\Theta}\left\|\frac{\partial a(\theta,Y_{-\ell'})}{\partial\theta_i}\right\|<\infty,\quad\tilde\PP^\thv\as$$
Thus, $\tilde\PP^\thv\as$, $\left\|\sum_{\ell'=0}^m\left(\prod_{\ell=0}^{m}a_{\ell', i}(\theta,Y_{-\ell})\right)z\right\|$ converges uniformly (in $\theta$) to zero. Moreover, there exist a constant $\rho_1\in(0,1)$ and a $\tilde\PP^\thv\as$ finite random variable $\tilde M_1>0$ with $\PEY[\tilde M_1]<\infty$ such that for all $m\in\zsetp$,
\begin{equation}\label{eq:ineq:diff:a:M:rho1}
\sum_{\ell'=0}^m\sup_{\theta\in\Theta}\left\|\prod_{\ell=0}^{m}a_{\ell', i}(\theta,Y_{-\ell})\right\|\le\tilde M_1\rho_1^m,\quad\tilde\PP^\thv\as
\end{equation}

Next, we show that $\tilde\PP^\thv\as$,
$$
\sum_{k=0}^m \left(\sum_{\ell'=0}^{k-1}\prod_{\ell=0}^{k-1}a_{\ell', i}(\theta,Y_{-\ell})\right)b(\theta,Y_{-k})
$$
converges uniformly on $\Theta$.  We obtain from \eqref{eq:ineq:diff:a:M:rho1}, $\tilde\PP^\thv\as$,
\begin{align*}
\sum_{k=0}^m\left\| \left(\sum_{\ell'=0}^{k-1}\prod_{\ell=0}^{k-1}a_{\ell', i}(\theta,Y_{-\ell})\right)b(\theta,Y_{-k})\right\| &\le\tilde M_1\sum_{k=0}^m\rho_1^{k-1}\sup_{\theta\in\Theta}\left\|b(\theta,Y_{-k})\right\| \\
&\le\tilde M_1\sum_{k=0}^\infty\rho_1^{k-1}\sup_{\theta\in\Theta}\left\|b(\theta,Y_{-k})\right\|,
\end{align*}
which is finite from \ref{assum:phi:linear:od}\ref{item:finite:sup:b:od} and \autoref{lem:useful:gen}.
To conclude \eqref{eq:def:first:diif:psi:infty:gen:od}, it remains to show that $\tilde\PP^\thv\as$,
$$
\sum_{k=0}^m\left(\prod_{\ell=0}^{k-1}a(\theta,Y_{-\ell})\right)\frac{\partial b(\theta,Y_{-k})}{\partial\theta_i}
$$
converges uniformly on $\Theta$.  We have $\tilde\PP^\thv\as$,
\begin{align*}
\sum_{k=0}^m\left\|\left(\prod_{\ell=0}^{k-1}a(\theta,Y_{-\ell})\right)\frac{\partial b(\theta,Y_{-k})}{\partial\theta_i}\right\|&\le\sum_{k=0}^mC\rho^{k}\sup_{\theta\in\Theta}\left\|\frac{\partial b(\theta,Y_{-k})}{\partial\theta_i}\right\|\\
&\le C\sum_{k=0}^\infty\rho^{k}\sup_{\theta\in\Theta}\left\|\frac{\partial b(\theta,Y_{-k})}{\partial\theta_i}\right\|,
\end{align*}
which is finite by \ref{assum:phi:linear:od}\ref{item:ineq:first:diff-a-b:finite:od} and \autoref{lem:useful:gen}. Thus the sequence $\nsequence{\frac{\partial\F[\theta]{\chunk{Y}{-m}{0}}(z)}{\partial\theta}}{m\in\zsetp}$ $\tilde\PP^\thv\as$ converges uniformly on $\Theta$ and this follows that  for all $z\in\Zset$ and $\theta\in\Theta$,
\begin{align*}
\lim_{m\to\infty}\frac{\partial\F[\theta]{\chunk{Y}{-m}{0}}(z)}{\partial\theta}=\frac{\partial}{\partial\theta}\lim_{m\to\infty}\F[\theta]{\chunk{Y}{-m}{0}}(z)=\frac{\partial\F[\theta]{\chunk{Y}{-\infty}{0}}}{\partial\theta},\; \tilde\PP^\thv\as,
\end{align*}
showing that the quantity $\frac{\partial\F[\theta]{\chunk{Y}{-\infty}{0}}}{\partial\theta}$ is $\tilde\PP^\thv\as$ well defined. The fact that \eqref{eq:finite:F2:gen:od} holds follows from \ref{assum:phi:linear:od}\ref{item:ineq:first:diff-a-b:finite:od}, \eqref{eq:finite:F1:gen:od} and \eqref{eq:ineq:diff:a:M:rho1}.

For any $i, j$, we have

\begin{align*}
\frac{\partial^2\F[\theta]{\chunk{Y}{-m}{0}}(z)}{\partial\theta_j\partial\theta_i} \\
&=\sum_{\ell'=0}^m\frac{\partial}{\partial\theta_j}\left(\prod_{\ell=0}^{m}a_{\ell', i}(\theta,Y_{-\ell})\right)z \\
&+ \sum_{k=0}^m \frac{\partial}{\partial\theta_j}\left[\left(\sum_{\ell'=0}^{k-1}\prod_{\ell=0}^{k-1}a_{\ell', i}(\theta,Y_{-\ell})\right)b(\theta,Y_{-k})\right] \\
&+\sum_{k=0}^m\frac{\partial}{\partial\theta_j}\left[\left(\prod_{\ell=0}^{k-1}a(\theta,Y_{-\ell})\right)\frac{\partial b(\theta,Y_{-k})}{\partial\theta_i}\right] \\
&=U_{1,m}(\theta)+U_{2,m}(\theta)+U_{3,m}(\theta)+U_{4,m}(\theta)+U_{5,m}(\theta),
\end{align*}
where
\begin{align*}
U_{1,m}(\theta)&=\sum_{\ell'=0}^m\sum_{\ell''=0}^m\left(\prod_{\ell=0}^{m}a_{\ell',\ell'',i,j}(\theta,Y_{-\ell})\right)z, \\
U_{2,m}(\theta)&=\sum_{k=0}^m\left(\sum_{\ell'=0}^{k-1}\sum_{\ell''=0}^{k-1}\prod_{\ell=0}^{k-1}a_{\ell',\ell'',i,j}(\theta,Y_{-\ell})\right)b(\theta,Y_{-k}), \\
U_{3,m}(\theta)&=\sum_{k=0}^m \left(\sum_{\ell'=0}^{k-1}\prod_{\ell=0}^{k-1}a_{\ell', i}(\theta,Y_{-\ell})\right)\frac{\partial b(\theta,Y_{-k})}{\partial\theta_j}, \\
U_{4,m}(\theta)&=\sum_{k=0}^m\left(\sum_{\ell'=0}^{k-1}\prod_{\ell=0}^{k-1}a_{\ell', j}(\theta,Y_{-\ell})\right)\frac{\partial b(\theta,Y_{-k})}{\partial\theta_i}, \\
U_{5,m}(\theta)&=\sum_{k=0}^m\left(\prod_{\ell=0}^{k-1}a(\theta,Y_{-\ell})\right)\frac{\partial^2 b(\theta,Y_{-k})}{\partial\theta_j\partial\theta_i},
\end{align*}
and with for all $i, j,\ell, \ell',\ell''$,
\begin{equation}
a_{\ell',\ell'',i,j}(\theta,Y_{-\ell})=\begin{cases}
 a_{\ell',i}(\theta,Y_{-\ell}) & \mbox{if}\quad\ell\neq\ell'' \\
\frac{\partial a_{\ell',i}(\theta,Y_{-\ell})}{\partial\theta_j} & \mbox{if}\quad\ell=\ell'',
\end{cases}
\end{equation}
and from~\eqref{eq:def:a:ell:i},
\begin{equation}\label{eq:deriv:a:ell:i}
\frac{\partial a_{\ell',i}(\theta,Y_{-\ell})}{\partial\theta_j}=\begin{cases}
 \frac{\partial a(\theta,Y_{-\ell})}{\partial\theta_j} & \mbox{if}\quad\ell\neq\ell' \\
\frac{\partial^2a(\theta,Y_{-\ell})}{\partial\theta_j\partial\theta_i} & \mbox{if}\quad\ell=\ell'.
\end{cases}
\end{equation}
To establish that $\nsequence{\frac{\partial^2\F[\theta]{\chunk{Y}{-m}{0}}(z)}{\partial\theta_j\partial\theta_i}}{m\in\zsetp}$ is almost surely uniformly convergent on $\Theta$, it requires proving that for all $i\in\{1,\ldots,5\}$, the sequences $\nsequence{U_{i,m}(\theta)}{m\in\zsetp}$ converges uniformly on $\Theta$.
We have
\begin{align*}
&\left\|U_{1,m}(\theta)\right\|=\left\|\sum_{\ell'=0}^m\sum_{\ell''=0}^m\left(\prod_{\ell=0}^{m}a_{\ell',\ell'',i,j}(\theta,Y_{-\ell})\right)z\right\| \\
&\le\sum_{\ell'=0}^m\sum_{\ell''=0}^m\left\|\prod_{\ell=0}^{m}a_{\ell',\ell'',i,j}(\theta,Y_{-\ell})\right\|\|z\| \\
&\stackrel{(1)}{\le}\tilde M_1^2\|z\|\sum_{\ell'=0}^m\sum_{\ell''=0}^m\rho_1^m\sup_{\theta\in\Theta}\left\|\frac{\partial a_{\ell',i}(\theta,Y_{-\ell''})}{\partial\theta_j}\right\|  \\
&\stackrel{(2)}{\le}\tilde M_1^2\|z\|\sum_{\ell''=0}^m\rho_1^m\left(m\sup_{\theta\in\Theta}\left\|\frac{\partial a(\theta,Y_{-\ell''})}{\partial\theta_j}\right\|+\sup_{\theta\in\Theta}\left\|\frac{\partial^2a(\theta,Y_{-\ell''})}{\partial\theta_j\partial\theta_i}\right\|\right),
\end{align*}
where $\stackrel{(1)}{\le}$ follows from \eqref{eq:ineq:diff:a:M:rho1} and the submultiplicativity of the matrix norm $\|\cdot\|$, and $\stackrel{(2)}{\le}$ follows from \eqref{eq:deriv:a:ell:i}. By assuming
\begin{equation*}
\tilde\PE^\thv\left[\lnp\sup_{\theta\in\Theta}\left\|\frac{\partial a(\theta,Y_{1})}{\partial\theta_i}\right\|\right]<\infty
\end{equation*}
and
\begin{equation*}
\tilde\PE^\thv\left[\lnp\sup_{\theta\in\Theta}\left\|\frac{\partial^2a(\theta,Y_{1})}{\partial\theta_j\partial\theta_i}\right\|\right]<\infty,
\end{equation*}
then there exist a constant $\rho_2\in(0,1)$ and a $\tilde\PP^\thv\as$ finite random variable $\tilde M_2>0$ with $\PEY[\tilde M_2]<\infty$ such that for all $m\in\zsetp$,

\begin{equation*}
\le\sum_{\ell'=0}^m\sum_{\ell''=0}^m\left\|\prod_{\ell=0}^{m}a_{\ell',\ell'',i,j}(\theta,Y_{-\ell})\right\|\le\tilde M_2\rho_2^m,\quad\tilde\PP^\thv\as
\end{equation*}
Consequently, for all $z\in\Zset$,
\begin{align*}%\label{eq:result:1:diff2}
\lim_{m\to\infty}\sup_{\theta\in\Theta}\left\|U_{1,m}(\theta)\right\|=0,\quad\tilde\PP^\thv\as
\end{align*}
Thus, $\tilde\PP^\thv\as$, the sequence $\nsequence{U_{1,m}(\theta)}{m\in\zsetp}$ converges to zero uniformly on $\Theta$.
Next, we have

\begin{align*}
 \left\|U_{2,m}(\theta)\right\|&=\sum_{k=0}^m\left\|\left(\sum_{\ell'=0}^{k-1}\sum_{\ell''=0}^{k-1}\prod_{\ell=0}^{k-1}a_{\ell',\ell'',i,j}(\theta,Y_{-\ell})\right)b(\theta,Y_{-k})\right\| \\
 &\le\sum_{k=0}^m\left(\sum_{\ell'=0}^{k-1}\sum_{\ell''=0}^{k-1}\left\|\prod_{\ell=0}^{k-1}a_{\ell',\ell'',i,j}(\theta,Y_{-\ell})\right\|\right)\left\|b(\theta,Y_{-k})\right\| \\
 &\le\tilde M_2\sum_{k=0}^m\rho^{k-1}\sup_{\theta\in\Theta}\left\|b(\theta,Y_{-k})\right\|.
\end{align*}
Since $\lim_{m\to\infty}\sum_{k=0}^m\rho^{k-1}\sup_{\theta\in\Theta}\left\|b(\theta,Y_{-k})\right\|<\infty$, then $\tilde\PP^\thv\as$, the sequence $\nsequence{U_{2,m}(\theta)}{m\in\zsetp}$ converges uniformly on $\Theta$. To show that $\nsequence{U_{3,m}(\theta)}{m\in\zsetp}$ and $\nsequence{U_{4,m}(\theta)}{m\in\zsetp}$ converge uniformly on $\Theta$, $\tilde\PP^\thv\as$, we proceed similarly as above and apply \eqref{eq:ineq:diff:a:M:rho1} and \ref{assum:phi:linear:od}\ref{item:ineq:first:diff-a-b:finite:od}.
Finally, by the same argument, the almost surely uniform convergence of $\nsequence{U_{5,m}(\theta)}{m\in\zsetp}$ follows from \ref{assum:phi:linear:od}\ref{item:ineq:prod:rho:n:od} and \ref{assum:phi:linear:od}\ref{item:ineq:second:diff-a-b:finite:od}. The $\tilde\PP^\thv\as$ uniform convergence of $\nsequence{\frac{\partial^2\F[\theta]{\chunk{Y}{-m}{0}}(z)}{\partial\theta_j\partial\theta_i}}{m\in\zsetp}$ on $\Theta$ then implies that $\theta\mapsto\frac{\partial^2\F[\theta]{\chunk{Y}{-\infty}{0}}}{\partial\theta_j\partial\theta_i}$ is $\tilde\PP^\thv\as$ well defined and
\begin{equation*}
\frac{\partial^2\F[\theta]{\chunk{Y}{-\infty}{0}}}{\partial\theta_j\partial\theta_i}=\lim_{m\to\infty}\frac{\partial^2\F[\theta]{\chunk{Y}{-m}{0}}(z)}{\partial\theta_j\partial\theta_i},\quad\tilde\PP^\thv\as,
\end{equation*}
showing \eqref{eq:def:second:diif:psi:infty:gen:od}. The proof for  \eqref{eq:finite:F3:gen:od} follows by similar arguments as for  \eqref{eq:finite:F1:gen:od} and  \eqref{eq:finite:F2:gen:od}.
\end{proof}

\subsection{Proof of~\autoref{thm:ergodicity:gen}}
\label{sec:ergodicity-proofs-gen:od}


The proof of this theorem follows similar lines of \cite{douc2015handy}. To proceed, let us first recall a more general set of conditions derived in
\cite{dou:kou:mou:2013} in which the following definition is required.

\begin{definition}\label{def:coupling:con-gen:od}
  Let $\bar H$ be a probability kernel from $\Zset^2$ to
  $\Ysigma^{\otimes 2}\otimes\mathcal{P}(\{0,1\})$ satisfying the following marginal conditions,
  for all $(z,z') \in \Zset^2$ and $B \in \Ysigma$,
\begin{equation}\label{eq:marginalConditionsH:gen}
  \begin{cases}
\bar H((z,z');B \times \Yset\times\{0,1\})=H(z;B)=G(\Proj{p}{z};B)\,, \\
\bar H((z,z');\Yset \times B\times\{0,1\} )=H(z';B)=G(\Proj{p}{z'};B) \eqsp,
  \end{cases}
\end{equation}
and such that the following coupling condition holds:
\begin{equation}\label{eq:coupling-conditionGbar-gen:od}
\bar H((z,z');\{(y,y)~:~y\in\Yset\}\times\{1\})=\bar H((z,z');\Yset^2\times\{1\}).
\end{equation}
Define the following quantities successively.
\begin{itemize}[label=--]
\item The trace measure of $\bar H((z,z');\cdot)$ on the set $\{(y,y)~:~y\in\Yset\}\times\{1\}$ is
  denoted by
\begin{equation*}%\label{def:Gcheck}
\check H((z,z');B)=\bar H((z,z');\{(y,y)~:~y\in B\}\times\{1\}),\quad B\in\Ysigma.
\end{equation*}
\item The probability kernel $\bar R$ from $(\Zset^2,\Zsigma^{\otimes 2})$ to
  $(\Zset^2\times\{0,1\},\Zsigma^{\otimes2}\otimes\mathcal{P}(\{0,1\}))$ is
  defined for all $z,z'\in\Zset^2$ and $A\in \Xsigma^{\otimes2}$ by
\begin{equation*}%\label{def:Rbar}
\bar R((z,z');A\times\{1\})=\int_{\Yset}
\1_A(\psi_{y}(z),\psi_{y}(z'))\;\check H((z,z');\rmd y).
\end{equation*}
 \item The measurable function $\alpha$ from $\Zset^2$ to $[0,1]$ is defined by
 \begin{equation}\label{eq:expressionAlpha-gen:od}
\alpha(z,z')=\bar R((z,z');\Zset^2 \times \{1\})
%=\check G((x,x');\Yset)
=\bar H((z,z');\Yset^2\times\{1\})\eqsp.
\end{equation}
\item The kernel $\hat R$ is defined for all $(z,z') \in \Zset^2$
  and $A \in \Zsigma^{\otimes 2}$ by
\begin{equation} \label{def:Rhat:gen:od}
\hat R((z,z'); A)=
\begin{cases}
\displaystyle\frac{\bar R((z,z');A \times \{1\})}{\alpha(z,z')}
&\text{ if $\alpha(z,z')>0$,}\\
0&\text{ otherwise.}
\end{cases}
 \end{equation}
\end{itemize}
\end{definition}
We can now introduce the so-called \emph{contracting condition} which yields ergodicity.
\begin{hyp}{AG}
\item \label{assum:asympStrgFeller:general:gen:od} There exists a kernel $\bar H$
  yielding $\alpha$ and $\hat R$ as in~\autoref{def:coupling:con-gen:od}, a measurable
  function $W: \Zset^2 \to [1,\infty)$ satisfying
  Conditions~\ref{assum:alpha-phi:gen}\ref{assum:definition-gamma-x:gen} and~\ref{assum:alpha-phi:gen}\ref{assum:hyp:1-alpha:W:gen} and real
  numbers $(D,\zeta_1,\zeta_2,\rho) \in (\rset_+)^3 \times (0,1)$ such that for
  all $(z,z') \in \Xset^2$ and, for all $n\geq1$,
\begin{align}
& {\hat R}^n((z,z'); \Zmet) \leq D \rho^n \Zmet(z,z') \label{eq:majo:d:gen:od},\\
& {\hat R}^n((z,z'); \Zmet\times W) \leq D \rho^n \Zmet^{\zeta_1}(z,z')W^{\zeta_2}(z,z'). \label{eq:majo:d:w:gen:od}
\end{align}
\end{hyp}
Under Conditions~\ref{ass-ergo-Zset:gen}, \ref{assum:weakFeller-V:gen}, \ref{assum:reachable:gen} and
\ref{assum:asympStrgFeller:general:gen:od} and by combining Theorem~6, Proposition~8
and Lemma~7 in~\cite{dou:kou:mou:2013}, we immediately obtain the following
result.
\begin{theorem} \label{thm:uniqueInvariantProbaMeasure:gen:od}
  Assume~\ref{ass-ergo-Zset:gen}, \ref{assum:weakFeller-V:gen},
  \ref{assum:reachable:gen} and \ref{assum:asympStrgFeller:general:gen:od}. Then the Markov
  kernel $K$ admits a unique invariant distribution $\pi$ and $\pi_1(\bar
  V)<\infty$ for any $\bar V:\Xset\to\rsetp$ such that $\bar V\lesssim V$.
\end{theorem}

Assumptions ~\ref{ass-ergo-Zset:gen}, \ref{assum:weakFeller-V:gen} and \ref{assum:reachable:gen} are quite usual and easy to
check.  The key point to obtain ergodicity is thus to construct $\bar H$
satisfying~\ref{assum:asympStrgFeller:general:gen:od}. For this, we can also rely on
the following result which is an adaption of \cite[Lemma~9]{dou:kou:mou:2013}.
\begin{lemma} \label{lem:practical:conditions:gen:od} %\rd{Check it carefully}
Assume that there exists $(\varrho, \beta, D_1, D_2, \ell, \ell') \in (0,1) \times \rset\times\rsetp\times\rsetp\times\zsetpnz\times\zsetpnz$ such that for all $(z,z')\in \Zset^2$,
\begin{align}
& \hat R\left((z,z'); \1_{\left(D_1\Zmet(z,z'),\infty\right)}\circ\Zmet\right)=0, \label{eq:contrac:gen:k} \\
& \hat R^\ell\left((z,z'); \1_{\left(\varrho\Zmet(z,z'),\infty\right)}\circ\Zmet\right)=0, \label{eq:contrac:gen:ell}
\end{align}
and
\begin{align}
& \hat RW \leq D_2W, \label{eq:driftCond:W:gen:k} \\
& \hat R^{\ell'}W \leq W +\beta. \label{eq:driftCond:W:gen:ell}
\end{align}
Then, \eqref{eq:majo:d:gen:od} and \eqref{eq:majo:d:w:gen:od} hold.
\end{lemma}

\begin{remark}
\autoref{lem:practical:conditions:gen:od} still holds if we replace \eqref{eq:driftCond:W:gen:k} and \eqref{eq:driftCond:W:gen:ell} by
\begin{align*}
\hat R^{\ell'+r}W \leq W +\beta, \quad r\in\{0,\ldots,\ell'-1\}.
\end{align*}
\end{remark}
\begin{proof}
Without loss of generality, we can assume that $\ell=\ell'=m$, otherwise we can take $m=\ell\vee\ell'$. For positive integer $n$, write $n=km+r$, where $r\in\{0,\ldots, m-1\}$ and $k\in\zsetp$. We have
\begin{align*}
{\hat R}^n\left((z,z'); \Zmet\right)&={\hat R}^{km+r}\left((z,z'); \Zmet\right)=\int{\hat R}^{km}\left((z,z');\rmd z_{1}\rmd z_{1}'\right){\hat R}^r\left((z_1,z_1'); \Zmet\right) \\
				&=\int{\hat R}^{km}\left((z,z');\rmd z_{1}\rmd z_{1}'\right){\hat R}^r\left((z_1,z_1'); \Zmet\1_{\left[0,D_1\Zmet(z_1,z_1')\right]}\circ\Zmet\right) \\
				&\le \int{\hat R}^{km}\left((z,z');\rmd z_{1}\rmd z_{1}'\right)D_1^r\Zmet(z_1,z_1') \\
				&\le D_1^r \int{\hat R}^{(k-1)m}\left((z,z');\rmd z_{2}\rmd z_{2}'\right){\hat R}^{m}\left((z_2,z_2');\rmd z_{1}\rmd z_{1}'\right)\Zmet(z_1,z_1')  \\
				&\le \varrho D_1^r \int{\hat R}^{(k-1)m}\left((z,z');\rmd z_{2}\rmd z_{2}'\right)\Zmet(z_2,z_2') \\
				& \cdots \\
				& \le \varrho^k D_1^r \Zmet(z,z')\le\varrho^{-1}\left(\bigvee_{r=0}^{m-1}D_1^r\right)\left({\varrho^{1/m}}\right)^n\Zmet(z,z')\eqsp.
\end{align*}
Similarly, we have
\begin{align*}
&{\hat R}^n((z,z'); \Zmet\times W)=\int{\hat R}^{km}\left((z,z');\rmd z_{1}\rmd z_{1}'\right){\hat R}^r\left((z_1,z_1'); \Zmet\times W\right) \\
			&\le \int{\hat R}^{km}((z,z');\rmd z_{1}\rmd z_{1}')D_1^rD_2^r\Zmet(z_1,z_1')W(z_1,z_1') \\
			&\le D_1^rD_2^r \int{\hat R}^{(k-1)m}((z,z');\rmd z_{2}\rmd z_{2}'){\hat R}^{m}((z_2,z_2');\rmd z_{1}\rmd z_{1}')\Zmet(z_1,z_1') W(z_1,z_1')  \\
			&\le \varrho D_1^rD_2^r \int{\hat R}^{(k-1)m}((z,z');\rmd z_{2}\rmd z_{2}')\Zmet(z_2,z_2')(W(z_2,z_2')+\beta) \\
			& \cdots \\
			& \le D_1^rD_2^r\varrho^k (W(z,z')+\beta k)\Zmet(z,z')\eqsp.
\end{align*}
Since for all $\varrho\in(0,1)$, $\lim_{k\to\infty}k\varrho^k=0$,  then there exists $\bar D>0$ and $\bar \varrho\in[\varrho,1)$ such that
\begin{align*}
{\hat R}^n((z,z'); \Zmet\times W)& \le \bar D{\bar\varrho}^{k-1} W(z,z')\Zmet(z,z')		\\
						& \le \bar D(\bar\varrho^{1/m})^n W(z,z')\Zmet(z,z')\eqsp.	
\end{align*}
Setting $D=\bar D\vee\left(\varrho^{-1}\left(\bigvee_{r=0}^{m-1}D_1^r\right)\right)$ and $\rho=\varrho^{1/m}\vee \bar\varrho^{1/m}$, we get \eqref{eq:majo:d:gen:od} and \eqref{eq:majo:d:w:gen:od}.
\end{proof}

Now we can prove that our set of conditions is sufficient.
\begin{proof}[Proof of \autoref{thm:ergodicity:gen}]
We only need to show that  \ref{assum:bound:rho:gen}
  and~\ref{assum:alpha-phi:gen} imply ~\ref{assum:asympStrgFeller:general:gen:od}. We preface our proof by the following lemma, which is adapted from \cite[Lemma~5]{douc2015handy}.

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}\label{lem:coupling:con:gen:od}
Assume \ref{assum:alpha-phi:gen}\ref{item:alpha-phi:3:gen}.
Then one can define a kernel $\bar H$ as in~\autoref{def:coupling:con-gen:od} with
the same $\alpha$ given in~\eqref{eq:expressionAlpha-gen:od}.
Moreover, the kernel $\hat R$ defined by~\eqref{def:Rhat:gen:od} satisfies, for all $(z,z')\in\Zset^2$ such that
$\alpha(z,z')>0$ and all measurable functions $f:\Zset^2\to\rsetp$,
\begin{equation}\label{eq:rho-hatRef:gen:od}
\hat R((z,z');f)= G(\phi(z,z');\tilde{f})\quad\text{with}\quad
\tilde{f}(y)=f\left(\Psi_{y}(z),\Psi_{y}(z')\right).
\end{equation}
\end{lemma}
Let us complete the proof of  \autoref{thm:ergodicity:gen} before proving this
lemma. By \autoref{lem:coupling:con:gen:od} and  \autoref{lem:practical:conditions:gen:od}, it
remains to check that \eqref{eq:contrac:gen:k} and \eqref{eq:driftCond:W:gen:k} hold for all $(z,z')\in\Zset^2$.
 Observe that by definition of $\hat R$, we have for all $n\in\zsetpnz$, $(z,z')\in\Zset^2$ and measurable function $f:\Zset^2\to\rset$,
 $$
 \hat R^n\left((z,z');f\right)=J_n\left((z,z');f\right)\eqsp.
 $$
So, condition~\ref{assum:alpha-phi:gen}\ref{assum:driftCond:W:new:gen:od} can be equivalently stated as:
there exist $m\in\zsetpnz$ and $D>0$ such that $$\displaystyle
 \sup_{(z,z')\in\Zset^2} \left( \hat R^m\left((z,z');W\right) - W(z,z')\right)<\infty$$ and
 $$\sup_{(z,z')\in\Zset^2} \left( \hat R\left((z,z');W\right) - DW(z,z')\right)<\infty.$$
Thus we can find $(\beta, D_2)\in\rset\times(0,\infty)$ such
that~\eqref{eq:driftCond:W:gen:k}  and \eqref{eq:driftCond:W:gen:ell}  hold for all $(z,z')\in\Zset^2$.

Now, let $(z,z')\in\Zset^2$.
When $z=z'$, then for all $n\in\zsetpnz$ and $\chunk{y}{1}{n}\in\Yset^n$, $\Zmet(\F[ ]{\chunk{y}{1}{n}}(z),\F[ ]{\chunk{y}{1}{n}}(z'))=0$, implying that~\eqref{eq:contrac:gen:k} and ~\eqref{eq:contrac:gen:ell} hold with any nonnegative $D_1$ and $\varrho$. For $z\neq z'$, let $D_1$ and $\varrho$ be defined respectively by
\begin{equation*}%\label{eq:D1}
D_1=\bigvee_{k=1}^{\ell-1}\sup_{\substack{(z,z',\chunk{y}{1}{k})\in\Zset^2\times\Yset\\z\neq z'}}\frac{\Zmet(\F[ ]{\chunk{y}{1}{k}}(z),\F[ ]{\chunk{y}{1}{k}}(z'))}{\Zmet(z,z')},
\end{equation*}
and
\begin{equation*}%\label{eq:rho}
\varrho=\sup_{\substack{(z,z',\chunk{y}{1}{\ell})\in\Zset^2\times\Yset\\z\neq z'}}\frac{\Zmet(\F[ ]{\chunk{y}{1}{\ell}}(z),\F[ ]{\chunk{y}{1}{\ell}}(z'))}{\Zmet(z,z')}.
\end{equation*}
By~\ref{assum:bound:rho:gen}, we have $D_1\in(0,\infty)$ and $\varrho\in(0,1)$. Therefore, Conditions ~\eqref{eq:contrac:gen:k} and~\eqref{eq:contrac:gen:ell} hold for all $(z,z')\in\Zset^2$ with the above $D_1$ and $\varrho$.
\end{proof}
We conclude this section with the postponed
\begin{proof}[Proof of~\autoref{lem:coupling:con:gen:od}]
Let $(z, z')\in\Zset^2$. We define $\bar H((z,z');\cdot)$ as the distribution
of $(Y, Y',U)$ drawn as follows.  We first draw a random variable $\bar Y$
taking values in $\Yset$ with density $g(\phi(z,z');\cdot)$ with respect to
$\nu$. Then we define $(Y,Y',U)$ by separating the two cases,
$\alpha(z,z')=1$ and $\alpha(z,z')<1$.
  \begin{enumerate}[label=$\bullet$]
  \item Suppose that $\alpha(z,z')=1$. Then
    from~\ref{assum:alpha-phi:gen}\ref{item:alpha-phi:3:gen}, we have
 $$
 H(z;\cdot)=H(z';\cdot)=G(\phi(z,z');\cdot)\eqsp.
 $$
 In this case, we set $(Y,Y',U)=(\bar{Y},\bar Y,1)$.
\item Suppose now that $\alpha(z,z')<1$. Then,
 using~\eqref{eq:con:min-g-phi-gen:od}, the functions
$$
(1-\alpha({z,z'}))^{-1}\left[g(\Proj{p}{z};\cdot)-\alpha({z,z'})g(\phi(z,z');\cdot)\right]
$$
and
$$(1-\alpha({z,z'}))^{-1}\left[g(\Proj{p}{z'};\cdot)-\alpha({z,z'})g(\phi(z,z');\cdot)\right]\eqsp,$$
are probability density functions with respect to $\nu$ and we let
$\Lambda$ and $\Lambda'$ be two independent random variables taking values
 in $\Yset$ drawn with these two density functions,
respectively. In this case we draw $U$ independently according to a Bernoulli variable with
mean $\alpha(z, z')$ and set
\[
(Y,Y')=
\begin{cases} (\bar Y, \bar Y)  \quad &\text{if $U=1$} \eqsp,\\
(\Lambda, \Lambda') \quad &\text{if $U=0$} \eqsp.
\end{cases}
\]
  \end{enumerate}
One can easily check that the so defined kernel $\bar H$ satisfies ~\eqref{eq:marginalConditionsH:gen} and~\eqref{eq:coupling-conditionGbar-gen:od}. Moreover, for all $(z,z')\in\Zset^2$,
$$
\bar H((z,z');\Yset^2\times\{1\})=\PP(U=1)=
  \alpha(z,z'),
$$
which
is compatible with~(\ref{eq:expressionAlpha-gen:od}).
The kernel $\hat R$ is defined by setting $\hat R((z,z');\cdot)$ as the
conditional distribution of $(Z,Z')=(\Psi_Y(z),\Psi_Y(z'))$ given that
$U=1$.
To complete the proof of~\autoref{lem:coupling:con:gen:od}, observe that for any
measurable $f:\Zset^2\to\rsetp$, we have, for all $(z,z')\in\Zset^2$ such that $\alpha(z,z')>0$,
\begin{align*}
\hat R((z,z');f)&=\PE\left[f(\Psi_Y(z),\Psi_{Y}(z'))\mid U=1\right] \\
            &=\PE\left[f(\Psi_{\bar Y}(z),\Psi_{\bar Y}(z'))\right]\\
            &= G(\phi(z,z');\tilde{f}),
\end{align*}
where $\tilde{f}(y)= f(\Psi_y(z),\Psi_y(z'))$ for all $y\in\Yset$.

\end{proof}

\subsection{Proof of \autoref{lem:spec:rad:pos:1}}\label{sec:proof:lem:spec:rad:pos:1}
If $a_i\ge 0$, $b_i\ge 0$, for all $i\in\{1,\ldots,p\}$, and $\sum_{i=1}^p(a_i+b_i)< 1$, it is trivial that $\radius(\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T)<1$. Otherwise, there would exist some $\lambda$ such that $|\lambda|^{-1}\le1$ and
\begin{align*}
\left|\sum_{i=1}^p(a_i+b_i)\lambda^{-i}\right|\le\sum_{i=1}^p(a_i+b_i)\left|\lambda\right|^{-i}<1.
\end{align*}
This would imply $P(\lambda)=\lambda^{2p-1}\left(1-\sum_{k=1}^p(a_k+b_k)\lambda^{-k}\right)\neq0$.
%then by Rouch\'{e}'s theorem, all the $p$ roots of $Q(\lambda)$ lie within the unit disc centered at the origin. Hence the spectral radius $\radius(\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T)$ is less than $1$.
Suppose on the other hand  that $\lambda^\star=\radius(\mathbf{A}+\mathbf{b}\boldsymbol{\delta}_p^T)<1$. From~\eqref{eq:P:char:poly:gen}, we can write $P(\lambda)$ as a product of $\lambda^{p-1}$ and $Q(\lambda)=\lambda^p-\sum_{k=1}^p(a_k+b_k)\lambda^{p-k}$. Note that $\lambda^\star=0$ if and only if $\sum_{i=1}^p(a_i+b_i)=0$. Then we can suppose that $0<\lambda^\star<1$, then there exists $j\in\{1,\ldots,p\}$ such that $ a_j+b_j>0$. Furthermore, note that for all $i\in\{1,\ldots,p\}$, $(a_i+b_i){\lambda^\star}^i\ge(a_i+b_i){\lambda^\star}^p$ and this inequality is strict when $i=j$. Thus from Perron-Frobenius theorem for nonnegative matrix, $Q(\lambda^\star)=0$, and we have\begin{equation*}
0<{\lambda^\star}^p\left(1-\sum_{i=1}^p(a_i+b_i)\right)\eqsp.
\end{equation*}
This completes the proof since $\lambda^\star>0$.

\subsection{Proof of~\autoref{lem:support:bound:moment:pois}}\label{sec:proof:lem:support:bound:moment:pois}
If $\vartheta\le0$, then \eqref{eq:lem:bound:moment:pois:gen1} holds since $(1+Y)^{\vartheta}\le1$. Note that for $s\in(0,1]$, we have
 $$
 \PE[(1+Y)^s]\le\left(\PE[1+Y]\right)^s\le(1+\zeta)^s\le1+\zeta^s.
 $$
 Thus, \eqref{eq:lem:bound:moment:pois:gen1} holds when $\vartheta\in(0,1]$. For $\vartheta\in(1,3/2]$, then there exists $s_1\in(0,1/2]$ such that $\vartheta=1+s_1$. By Cauchy-Schwartz inequality, we have
\begin{align*}
 \PE[(1+Y)^\vartheta]&=\PE[(1+Y)(1+Y)^{s_1}]\le \left(\PE[(1+Y)^{2}]\right)^{1/2} \left(\PE[(1+Y)^{2s_1}]\right)^{1/2} \\
 &\le(1+3\zeta+\zeta^2)^{1/2}(1+\zeta)^{s_1}\le c_1+c_2\zeta^{\vartheta},
\end{align*}
 for some constants $c_1, c_2>0$.
If $\vartheta\in(3/2,2]$, then there exists $s_2\in(0,1/2]$ such that $\vartheta=3/2+s_2$ and
\begin{align*}
 \PE[(1+Y)^\vartheta]&=\PE[(1+Y)^{3/2}(1+Y)^{s}]\le \left(\PE[(1+Y)^{3}]\right)^{1/2} \left(\PE[(1+Y)^{2s_2}]\right)^{1/2} \\
 &\le(1+7\zeta+6\zeta^2+\zeta^3)^{1/2}(1+\zeta)^{s_2}\le c_3+c_4\zeta^{\vartheta},
\end{align*}
 for some constants $c_3, c_4>0$. Finally, if $\vartheta>2$, then there exists $s_3\in(0,1]$ such that $\vartheta=\lfloor\vartheta\rfloor+s_3$ and we have
\begin{align*}
 \PE\left[(1+Y)^\vartheta\right]&=\PE\left[(1+Y)^{\lfloor\vartheta\rfloor}(1+Y)^{s_3}\right]\\
 &\le \left(\PE\left[(1+Y)^{2\lfloor\vartheta\rfloor}\right]\right)^{1/2} \left(\PE\left[(1+Y)^{2s_3}\right]\right)^{1/2} \\
 &\le\left(1+2^{\lfloor\vartheta\rfloor}\left(\PE\left[Y^{2\lfloor\vartheta\rfloor}\right]\right)^{1/2}\right)\left(c_5+c_6\zeta^{s_3}\right)\eqsp,
\end{align*}
 for some constants $c_5, c_6>0$. The proof then follows by observing that
$$
\lim_{\zeta\to\infty}\frac{\left(\PE\left[Y^{2\lfloor\vartheta\rfloor}\right]\right)^{1/2}}{\zeta^{\lfloor\vartheta\rfloor}}=1\eqsp.
$$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{lemma} \label{lem:useful:gen}
Let \sequence{U}[k][\zsetp] be a stationary sequence of real-valued random variables on $(\Omega, \mcf, \PP)$. Assume that $\PE{(\lnp |U_0|)}<\infty$. Then, for all $\eta \in (0,1)$,
$$
\lim_{k \to \infty} \eta^k U_k =0\, , \quad \PP\as
$$
\end{lemma}
\begin{proof}
See~\cite[Lemma~34]{dou:kou:mou:2013}.
\end{proof}





%\begin{appendix}\label{sec:appendix}
%
%%\subsection{Proof of \autoref{lem:imp:p;seq}}
%%\label{sec:proof-lem:imp}
%%First observe that,  by induction on $n$, having ~\eqref{eq:sub:lem:p}
%%for all $n\geq2$ is equivalent to having, for all $n\geq2$,
%%\begin{multline*}
%%p^{\theta,\thv} (\chunk{Y}{1}{n}|\chunk{Y}{-\infty}{0}) \\
%%=p^{\theta,\thv}(Y_{n}|\chunk{Y}{-\infty}{n-1})
%%p^{\theta,\thv}(Y_{n-1}|Y_{-\infty:n-2})\cdots p^{\theta,\thv} (Y_{1}|\chunk{Y}{-\infty}{0}),\ \tilde{\PP}^{\thv}\as,
%%\end{multline*}
%%which, using that $\tilde{\PP}^{\thv}$ is shift-invariant,
%%is in turn equivalent to having that, for all $n\geq2$,
%%\begin{equation}\label{eq:sub:lem:p2}
%%p^{\theta,\thv} (\chunk{Y}{1}{n}|\chunk{Y}{-\infty}{0})
%%=p^{\theta,\thv} (\chunk{Y}{2}{n}|\chunk{Y}{-\infty}{1})p^{\theta,\thv} (Y_{1}|\chunk{Y}{-\infty}{0}), \quad \tilde{\PP}^{\thv}\as
%%\end{equation}
%%Thus to conclude the proof, we only need to show that~(\ref{eq:sub:lem:p2})
%%holds for all $n\geq2$. By \autoref{def:cond-densities-given-past}, we have, for all
%%$n\geq2$, and all $y_{-\infty:n}\in\Yset^{\zset_-}$,
%%\begin{multline*}
%%p^{\theta,\thv} (\chunk{y}{2}{n}|\chunk{y}{-\infty}{1})p^{\theta,\thv} (y_{1}|\chunk{y}{-\infty}{0})\\
%%=\int\Phi^{\theta,\thv}(\chunk{y}{-\infty}{1};\rmd x_1)p^{\theta,\thv} (y_{1}|\chunk{y}{-\infty}{0})\prod_{k=1}^{n-1} \kap{y_{k}, y_{k+1}}{x_{k}; \rmd x_{k+1}}\eqsp.
%%\end{multline*}
%%Now, using~\ref{assum:exist:phi:theta}, we get that, for all $n\geq2$,
%%\begin{align*}
%%&p^{\theta,\thv} (\chunk{Y}{2}{n}|\chunk{Y}{-\infty}{1})p^{\theta,\thv} (Y_{1}|\chunk{Y}{-\infty}{0})\\
%%&=\int\Phi^{\theta,\thv}(\chunk{Y}{-\infty}{0};\rmd x_0)\prod_{k=0}^{n-1} \kap{y_{k}, y_{k+1}}{x_{k}; \rmd x_{k+1}}, \quad \tilde{\PP}^{\thv}\as \\
%%&=p^{\theta,\thv} (\chunk{Y}{1}{n}|\chunk{Y}{-\infty}{0}), \quad \tilde{\PP}^{\thv}\as \;,
%%\end{align*}
%%which yields~(\ref{eq:sub:lem:p2}). Thus the proof follows.
%
%
%
%
%\section{Proof of~\autoref{lem:exist:practical-cond:O}.}\label{app:lem}
%Under~\ref{ass:21-lyapunov:gen:od},
%Assumptions~\ref{assum:practical-cond:O:gen:od}\ref{item:22or23-barphi-V:gen:od} implies that for all $\theta\in\Theta$,
%\begin{equation}
%\label{eq:bar:phi:mc}
%\pi^\theta_2\left(\lnp (\bar{\phi})\right)<\infty\;,
%\end{equation}
%and if moreover $C>0$,
%\begin{equation}
%\label{eq:bar:phi:strong:mc}  \pi^\theta_2\left(\bar{\phi}\right)<\infty \;.
%\end{equation}
%For proving~\autoref{lem:exist:practical-cond:O}, we will also make use of
%\cite[Lemma~34]{dou:kou:mou:2013} which we quote here for convenience.
%\begin{lemma} \label{lem:useful:gen}
%Let \sequence{U}[n][\zsetp] be a stationary sequence of real-valued random variables on $(\Omega, \mcf, \PP)$. Assume that $\PE{(\lnp |U_0|)}<\infty$. Then, for all $\eta \in (0,1)$,
%$$
%\lim_{k \to \infty} \eta^k U_k =0\, , \quad \PP\as
%$$
%\end{lemma}
%\begin{proof}[Proof of~\autoref{lem:exist:practical-cond:O}]
%  We first show that $p^{\theta}(y|\chunk{Y}{-\infty}{0})$
%  in~(\ref{eq:def-p-theta-neq-thv:gen:od}) is finite for $x=x_1$ $\tilde \PP^{\thv}\as$\ By~\ref{assum:continuity:Y:g-theta:gen:od}, this follows
%by writing
%\begin{equation}\label{eq:p-theta-thv:def}
%p^{\theta}\left(y_1\,|\,\chunk{y}{-\infty}{0}\right) =
%g^\theta\left(\F[\theta]{\chunk{y}{-\infty}{0}};y_1\right)\eqsp,
%\end{equation}
%if, for all $\theta,\thv\in\Theta$, the limit
%\begin{equation}
%  \label{eq::exist:x0:O}
%\F[\theta]{\chunk{Y}{-\infty}{0}}=\lim_{m \to \infty}\f{\chunk{Y}{-m}{0}}(x_1)
%\quad\text{ is well defined}\quad \tilde \PP^{\thv}\as
%\end{equation}
%For all $\theta\in\Theta$, $m\geq 0$,
%  $x\in\Xset$ and $\chunk{y}{-m}{0}\in\Yset^{m+1}$,
%  using~\ref{assum:practical-cond:O:gen:od}\ref{assum:exist:practical-cond:O:gen:od}, we have
%\begin{equation} \label{eq:diff:x:y}
%\Xmet(\F[\theta]{\chunk{y}{-m}{0}}(x_1),\F[\theta]{\chunk{y}{-m}{0}}(x)) \leq \varrho^{m+1} \; \bar\psi(x)\eqsp.
%\end{equation}
%Taking $x=\psi^\theta_{y_{-m-1}}(x_1)$ and
%using~\ref{assum:practical-cond:O:gen:od}\ref{item:barphi:bar:phi:gen:od}, we obtain, for
%all  $\chunk{y}{-m-1}{0}\in\Yset^{m+2}$,
%\begin{equation*}
%\Xmet(\F[\theta]{\chunk{y}{-m}{0}}(x_1),\F[\theta]{\chunk{y}{-m-1}{0}}(x_1)) \leq \varrho^{m+1} \; \bar\phi\left(y_{-m-1}\right)\eqsp.
%\end{equation*}
%Using~(\ref{eq:bar:phi:mc}) and
%\autoref{lem:useful:gen}, we have that
%\begin{equation}
%  \label{eq:phibar-infty}
%\forall \eta\in(0,1),\quad\sum_{k\in\zset}\eta^{|k|}  \bar\phi\left(Y_{k}\right) < \infty\eqsp,\quad\tilde\PP^{\thv}\as\eqsp,
%\end{equation}
%and thus $\left(\F[\theta]{\chunk{Y}{-m}{0}}(x_1)\right)_{m\geq0}$ is a Cauchy
%sequence $\tilde\PP^{\thv}\as$ Its limit exists $\tilde\PP^{\thv}\as$, since
%$(\Xset,d)$ is assumed to be complete, which defines the $\Xset$-valued
%random variable $\F[\theta]{\chunk{Y}{-\infty}{0}}$ for all
%$\theta,\thv\in\Theta$ when $Y$ has distribution  $\tilde\PP^{\thv}\as$ Thus~(\ref{eq::exist:x0:O}) holds and we further obtain that
%\begin{align}\nonumber
% \sup_{\theta\in\Theta}
% \Xmet(\F[\theta]{\chunk{Y}{-k}{0}}(x_1),x_1)
%&\leq  \sup_{\theta\in\Theta}\sum_{m=0}^{k}
% \Xmet(\F[\theta]{\chunk{Y}{-m}{0}}(x_1),\F[\theta]{\chunk{Y}{-m+1}{0}}(x_1))\\
%  \label{eq:limit-bound-cauchy-beforelimit}
%&\leq\sum_{m\ge0}\varrho^{m} \;
% \bar\phi\left(Y_{-m}\right)<\infty\eqsp, \quad\tilde\PP^{\thv}\as
%\end{align}
%so that, letting $k\to\infty$,
%\begin{align}
%  \label{eq:limit-bound-cauchy}
% \sup_{\theta\in\Theta}
% \Xmet(\F[\theta]{\chunk{Y}{-\infty}{0}},x_1)
%\leq\sum_{m\ge0}\varrho^{m} \;
% \bar\phi\left(Y_{-m}\right)<\infty\eqsp, \quad\tilde\PP^{\thv}\as
%\end{align}
%Let us now
%prove~\ref{assum:Ptheta:thetastar:is:density:gen:od}. Relation~(\ref{eq:p-theta-thv:def})
%directly
%yields~\ref{assum:Ptheta:thetastar:is:density:gen:od}\ref{item:PthetaNEQthetastar:is:density:gen:od}.
%Let us
%prove~\ref{assum:Ptheta:thetastar:is:density:gen:od}\ref{item:PthetaISthetastar:is:density:gen:od},
%hence consider the case $\theta=\thv$. Using~\eqref{eq:diff:x:y}, we have
%\begin{equation*}
%\Xmet(\F[\thv]{\chunk{Y}{-m}{0}}(x_1),\F[\thv]{\chunk{Y}{-m}{0}}(X_{-m})) \leq \varrho^{m+1} \; \bar\psi(X_{-m})\quad\PP^{\thv}\as
%\end{equation*}
%Since $\{\bar\psi(X_{-m})\}_{m\geq0}$ is stationary under $\PP^\thv$,
%it is bounded in probability, and since $\varrho<1$, for all $\epsilon>0$, we have
%\begin{equation}
%  \label{eq:proba-limit}
% \lim_{m\to\infty}\PP^{\thv}\left(\Xmet\left(\F[\thv]{\chunk{Y}{-m}{0}}(X_{-m}),\F[\thv]{\chunk{Y}{-m}{0}}(x)\right)>\epsilon\right)=0\eqsp.
%\end{equation}
%Note that for all $m\geq1$, $\F[\thv]{\chunk{Y}{-m}{0}}(X_{-m})=X_1$ $\PP^{\thv}\as$, hence
%we get that
%\begin{equation}
%  \label{eq:eq-ftheta'theta'X1}
%\F[\thv]{\chunk{Y}{-\infty}{0}}= X_1 \quad  \PP^{\thv}\as
%\end{equation}
%To complete the proof of~\ref{assum:Ptheta:thetastar:is:density:gen:od}\ref{item:PthetaISthetastar:is:density:gen:od}, we need to show that, under $\tilde\PP^{\thv}$, $y\mapsto g^\thv
%(\F[\thv]{\chunk{Y}{-\infty}{0}};y)=g^\thv
%(X_1;y)$ is the conditional density of
%$Y_1$ given $\chunk{Y}{-\infty}{0}$, that is, for any $B\in\Ysigma$,
%$$
%\int\1_{B}(y)g^{\thv}\left(X_1;y\right)\nu(\rmd
%y)=\PP^{\thv}\left(Y_1\in B\,|\,\chunk{Y}{-\infty}{0}\right) \;.
%$$
%Now, note that, by defintion of $\PP^{\thv}$,
%$$
%\int\1_{B}(y)g^\thv\left(X_1;y\right)\nu(\rmd y) = \PP^{\thv}
%\left(Y_1\in B\mid X_1\right) =\PP^{\thv} \left(Y_1\in B\mid X_1,\chunk{Y}{-\infty}{0}\right)\;.
%$$
%But since~(\ref{eq:eq-ftheta'theta'X1}) implies that $X_1$ is
%$\sigma(\chunk{Y}{-\infty}{0})$-measurable, $X_1$ can be removed in the last
%conditioning, which concludes the proof~\ref{assum:Ptheta:thetastar:is:density:gen:od}\ref{item:PthetaISthetastar:is:density:gen:od}.
%
%
%Finally, it remains to show the uniform
%convergence~(\ref{eq:unfi-limit-contrast-od}) in~\ref{assum:limit:Y}.
%By~\ref{assum:continuity:Y:phi-theta:gen:od} and~(\ref{eq::exist:x0:O}), we have, for
%all $\theta,\thv\in\Theta$, $k\in\zsetp$,
%\begin{equation}\label{eq:com:psi:inf}
%\F[\theta]{\chunk{Y}{-\infty}{k-1}}=\F[\theta]{\chunk{Y}{1}{k-1}}\left(\F[\theta]{\chunk{Y}{-\infty}{0}}\right)\eqsp,\quad\tilde\PP^{\thv}\as
%\end{equation}
%From~\ref{assum:practical-cond:O:gen:od}\ref{assum:exist:practical-cond:O:gen:od} and
%\eqref{eq:com:psi:inf}, we get
%$$
%\Xmet(\F[\theta]{\chunk{Y}{1}{k-1}}(x_1),\F[\theta]{\chunk{Y}{-\infty}{k-1}})
%\le\varrho^{k-1} \bar\psi\left(\F[\theta]{\chunk{Y}{-\infty}{0}}\right)\eqsp,\quad\tilde\PP^{\thv} \as
%$$
%On the other hand~\ref{assum:practical-cond:O:gen:od}\ref{item:psibar:gen:od}
%and~(\ref{eq:limit-bound-cauchy}) imply
%\begin{equation}\label{eq:finite:sup}
%\sup_{\theta\in\Theta}\bar\psi\left(\F[\theta]{\chunk{Y}{-\infty}{0}}\right)<\infty\eqsp,\quad\tilde\PP^{\thv}\as\eqsp,
%\end{equation}
%which, with the previous display, yields,
%\begin{equation}\label{eq:A:sup:theta}
%\sup_{\theta\in\Theta}
%\Xmet(\F[\theta]{\chunk{Y}{1}{k-1}}(x_1),\F[\theta]{\chunk{Y}{-\infty}{k-1}})
%=O_{k\to\infty}\left(\varrho^{k}\right)\quad\tilde\PP^{\thv}\as
%\end{equation}
%Since $\Xset_1$ is closed and satisfies
%Condition~\ref{assum:practical-cond:O:gen:od}\ref{assum:exist:practical-cond:subX:gen:od},
%we have that, $\f{\chunk{Y}{1}{k-1}}(x_1)$ and
%$\F[\theta]{\chunk{Y}{-\infty}{k-1}}$ are in $\Xset_1$ for all $k\geq2$. Thus
%Condition~\ref{assum:practical-cond:O:gen:od}\ref{item:ineq:loggg:gen:od} gives that
%$$
%\sup_{\theta \in \Theta} \left|\ln \frac{g^\theta
%    (\f{\chunk{Y}{1}{k-1}}(x_1);Y_k)}{g^\theta
%    (\F[\theta]{\chunk{Y}{-\infty}{k-1}};Y_k)}\right|\leq A_k(1)\times
%A_k(2)\times A_k(3)\times A_k(4)\quad \tilde\PP^{\thv}\as\;,
%$$
%where
%\begin{align*}
%A_k(1)&=\sup_{\theta \in \Theta}
%H\left(\Xmet(\f{\chunk{Y}{1}{k-1}}(x_1),\F[\theta]{\chunk{Y}{-\infty}{k-1}})\right)\\
%A_k(2)&=\sup_{\theta \in \Theta}
%\rme^{C\,\Xmet\left(x_1,\F[\theta]{\chunk{Y}{-\infty}{k-1}}\right)}\\
%A_k(3)&=\sup_{\theta \in \Theta}
%\rme^{C\,\Xmet\left(x_1,\f{\chunk{Y}{1}{k-1}}(x_1)\right)}\\
%A_k(4)&=\bar\phi(Y_k) \;.
%\end{align*}
%By~\eqref{eq:A:sup:theta} and~\ref{assum:practical-cond:O:gen:od}\ref{item:H}, we
%have
%$$
%A_k(1)=O_{k\to\infty}\left(\varrho^{k}\right)\quad\tilde\PP^{\thv}\as
%$$
%With~(\ref{eq:phibar-infty}), this yields~(\ref{eq:unfi-limit-contrast-od}) in the case where $C=0$.
%For $C>0$, we further observe that, by~\eqref{eq:limit-bound-cauchy} and~(\ref{eq:bar:phi:strong:mc}), we have, for all $\thv\in\Theta$ and $k\in\zsetp$,
%\begin{align*}
%\tilde\PE^{\thv}\left[\lnp A_k(2) \right]
%\le\tilde\PE^{\thv}\left[C\sum_{m\ge0}^\infty\varrho^m\bar\phi\left(Y_{-m+k-1}\right)\right]=\frac{C\pi_2^{\thv}\left(\bar\phi\right)}{1-\varrho}<\infty\eqsp.
%\end{align*}
%Then~\autoref{lem:useful:gen} implies that, $\tilde\PP^{\thv}\as$,
%$A_k(2)=O(\eta^{-k})$ for any $\eta\in(0,1)$. The same property applies
%similarly to $A_k(3)$ by using~(\ref{eq:limit-bound-cauchy-beforelimit}) in
%place of~(\ref{eq:limit-bound-cauchy}). This yields~(\ref{eq:unfi-limit-contrast-od}) in
%the case where $C>0$, which concludes the proof.
%\end{proof}
%
%\end{appendix}
%
%\begin{center}
%\section*{Acknowledgement}
%\end{center}
%We are thankful to the editor-in-charge and anonymous referee for the insightful comments and the helpful suggestions that lead to improve this paper.

%\




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