\subsection{NBIN-GARCH model}

\autoref{example:Nbigarch:defi} is a specific case of \autoref{def:obs-driv:gen} where $\nu$ is the counting measure on $\Yset=\nset$,
\begin{align}
&\psi^\theta_y(x)=\omega+ax+by\eqsp,\label{eq:Nbingarch:def:phi}\\
&g^\theta(x;y)
=\frac{\Gamma(y+r)}{y!\Gamma(r)}\left(\frac{1}{1+x}\right)^r\left(\frac{x}{1+x}\right)^y \eqsp,\label{eq:Nbingarch:g}
\end{align}
with $\theta=(\omega, a, b, r)$ in a compact subset $\Theta$ of
$(0,\infty)^2\times\rsetp^2$ and $\Xset=(0,\infty)$.


In \cite[Theorem~1]{zhu:2011},
the equation satisfied by the mean of the observations $\mu_k=\PE[Y_k]$ is
derived and is shown to admit a constant solution if and only if
\begin{equation}
  \label{eq:nbingarch-stability-cond-zhu}
  rb+a<1 \;.
\end{equation}
This clearly implies that this condition is necessary to have a stationary
solution $\{Y_k\}$ with finite mean. However it does not imply the existence of
such a solution. In fact, the following result shows
that~(\ref{eq:nbingarch-stability-cond-zhu}) is indeed a necessary and
sufficient condition to have a stationary solution $\{Y_k\}$ with finite
mean. It also shows that all the assumptions of \autoref{thm:convergence-main}
hold, which, with \autoref{rem:identifiability}, provides the consistency of the
MLE $\mlY{x_1,n}$ for any  $x_1\in\Xset$.

\begin{theorem}\label{theo:ergo-converge:nbigarch}
  Suppose that all $\theta=(\omega, a, b, r)$ in $\Theta$ satisfy
  Condition~\eqref{eq:nbingarch-stability-cond-zhu}. Then
  Assumptions~\ref{assum:gen:identif:unique:pi}, \ref{ass:21-lyapunov},
  \ref{assum:continuity:Y:g-theta}, \ref{assum:continuity:Y:phi-theta} and \ref{assum:practical-cond:O} hold
  with $\bar V$ being defined as the identity function on $\Xset$ and with any $x_1\in\Xset$.
\end{theorem}


\begin{proof} For convenience, we divide the proof in two
  steps.\\
\noindent\textbf{Step 1.} We first prove
Assumptions~\ref{assum:gen:identif:unique:pi} and \ref{ass:21-lyapunov} by
applying \autoref{thm:ergodicity}. We set $\bar V(x)= V(x)=x$ and thus we only need to
check~~\ref{ass-ergo-Xset}, \ref{assum:weakFeller-V}, \ref{assum:weakFeller}, \ref{assum:reachable},
\ref{assum:bound:rho} and~\ref{assum:38--40:alpha-phi}. Condition~\ref{ass-ergo-Xset} holds. We have for all $\theta\in\Theta$,
$$
R V(x)=\omega + (a+br)x =(a+br)V(x) + \omega,
$$
which yields  \ref{assum:weakFeller-V}.
The fact that the kernel $R$ is weak Feller easily
follows by observing that, as $p\to p'$, $\mathcal{NB}(r,p)$ converges weakly
to $\mathcal{NB}(r,p')$, so \ref{assum:weakFeller} holds.

We now prove \ref{assum:reachable}. 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-Px}), 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} holds.
Assumption~\ref{assum:bound:rho} 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:38--40:alpha-phi}, we apply ~\autoref{lem:det:alpha:gen}
with $\Cset=\Xset$, $\Sset=\{1\}$ (so $\mu$ boils down to the Dirac measure on $\{1\}$). For all
$(x,y)\in\Xset\times\Yset$, let $j(x)=\left(\frac{1}{1+x}\right)^r$ and
$h(x;y)=\frac{\Gamma(y+r)}{y!\Gamma(r)}\left(\frac{x}{1+x}\right)^y$. Indeed,
$j$ is one-to-one and $h$ satisfies ~\ref{item:F1}. Thus
by~\autoref{lem:det:alpha:gen}, for all $(x,x')\in\Xset^2$ and $y\in\Yset$, we
get that
\begin{equation*}\label{eq:alpha:x:x'}
\alpha(x, x')=\left(\frac{1+x \wedge x'}{1+x \vee x'}\right)^r\in(0,1] \quad \text{and} \quad \phi(x,x')=x\wedge x'
\end{equation*}
satisfy~\ref{assum:38--40:alpha-phi}\ref{item:alpha:phi:1},
\ref{assum:38--40:alpha-phi}\ref{item:alpha:phi:2} and
\ref{assum:38--40:alpha-phi}\ref{item:alpha:phi:3}.
%According to~\autoref{lem:coupling:con}, $\bar G$ can be constructed. Next, we show that the constructed kernel $\bar G$ satisfies the {\color{blue}{contracting condition}} by proving that ~\eqref{eq:G:C:neq}--\eqref{eq:definition-gamma-x} hold. Clearly, ~\eqref{eq:G:C:neq} holds.
For any given $r>0$,  let a function $W\,:\,\Xset^2\to[1,\infty)$  be defined by, for all $(x,x')\in\Xset^2,\, W(x,x')=1\vee r$. By definition of $W$, as a constant function, \ref{assum:38--40:alpha-phi}\ref{assum:definition-gamma-x} and \ref{assum:38--40:alpha-phi}\ref{assum:driftCond:W:new}  clearly hold. Moreover, \ref{assum:38--40:alpha-phi}\ref{assum:hyp:beta:gene} holds since
for all $(x, x') \in\Xset^2$, we have that
\begin{equation*}
1-\alpha(x,x')\leq (1\vee r)\lvert x-x'\rvert = W(x,x')\lvert x-x'\rvert\eqsp.
\end{equation*}
Therefore, \ref{assum:38--40:alpha-phi} holds, which completes \textbf{Step 1}.

\noindent\textbf{Step 2}. We now prove  %\ref{assum:momentCond},
  \ref{assum:continuity:Y:g-theta}, \ref{assum:continuity:Y:phi-theta} and \ref{assum:practical-cond:O}.
By assumption on $\Theta$, then there exists $(\underline{\omega},\bar{\omega},\underline{b}, \bar b,\underline{r}, \bar r,\underline{\alpha}, \bar \alpha) \in (0,\infty)^6\times(0,1)^2$ such that
$$
\underline{\omega}\leq \omega \leq\bar \omega, \,\, \leq b \leq\bar{b}, \,\, \underline{r}\leq r\leq \bar{r}, \,\, \underline{\alpha}\leq a+br\leq \bar{\alpha}\eqsp.
$$
Clearly, \ref{assum:continuity:Y:g-theta} and
\ref{assum:continuity:Y:phi-theta} hold by definitions of $\psi^\theta_y(x)$
and $g^\theta(x;y)$.  It remains to check~\ref{assum:practical-cond:O} 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}\ref{assum:exist:practical-cond:subX} 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}\ref{assum:momentCond}.
From~\eqref{eq:notationItere:f:cl} and ~\eqref{eq:Nbingarch: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:nbingarch}
\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:nbingarch}, we have, for all $\theta\in\Theta$, $x\in\Xset$ and $y_{1:n}\in\Yset^n$,
\begin{equation*}%\label{eq:nbingarch:psi:x:x'}
\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}\ref{assum:exist:practical-cond:O}
  and~\ref{assum:practical-cond:O}\ref{item:psibar} 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}\ref{item:barphi:bar:phi}
  and~\ref{assum:practical-cond:O}\ref{item:ineq:loggg} and~\ref{assum:practical-cond:O}\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|(r+y)\left[\ln(1+x')-\ln(1+x)\right]+y\left[\ln x-\ln x'\right]\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*}
% &\left(r\left|\frac{\ln(1+x)-\ln(1+x')}{x-x'}\right|+y\left|\frac{\ln\left(\frac{x}{1+x}\right)-\ln\left(\frac{x'}{1+x'}\right)}{x-x'}\right|\right)|x-x'|\\
% &\leq \left(\bar r\frac{1}{1+\underline \omega}+y\frac{1}{\underline \omega(1+\underline \omega)}\right)|x-x'|\le\left({\bar r}+(1+\bar{\alpha})x_1+{\underline \omega}^{-1}y\right)|x-x'|\eqsp.
% \end{align*}
Setting  $\bar\phi(y)=\bar\omega\vee{\bar r}+(1+\bar{\alpha})x_1+\left(\bar{b}\vee(1+\underline \omega^{-1})\right)y$,
$H(x)=x$ and $C=0$ then yield Conditions~\ref{assum:practical-cond:O}\ref{item:barphi:bar:phi}, ~\ref{assum:practical-cond:O}\ref{item:ineq:loggg}
  and~\ref{assum:practical-cond:O}\ref{item:H}. Now
  \ref{assum:practical-cond:O}\ref{item:22or23-barphi-V} follows from
$$
\int \lnp y \; G^\theta(x,\rmd y) \leq \int y \; G^\theta(x,\rmd y) = r x\leq
\overline r \bar V(x)\;.
$$
This concludes the proof.
\end{proof}

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