\subsection{NM-GARCH model}

The NM$(d)$-GARCH$(1,1)$ of~\autoref{example:nmgarch:defi} is a specific case of \autoref{def:obs-driv:gen} where $\Xset=\rsetp^d$ and $\nu$ is the Lebesgue measure on $\Yset=\rset$,

\begin{align}
&\psi^\theta_y(\mathbf{x})=\boldsymbol{\omega}+\mathbf{A}\mathbf{x}+y^2\mathbf{b} \eqsp,
\label{eq:psi:nm-garch}\\
&g^\theta(\mathbf{x};y)= \sum_{\ell=1}^{d}{\gamma_\ell\frac{\rme^{-y^2/{2x_{\ell}}}}{(2\pi x_{\ell})^{1/2}}}\eqsp, \quad (\mathbf{x},y)\in\Xset\times\Yset\eqsp,
\label{eq:g-theta:nm-garch}
\end{align}
and $\theta=\left(\boldsymbol{\gamma}, \boldsymbol{\omega}, \mathbf{A},
  \mathbf{b}\right)\in\Theta$, a compact subset of
$\simplex_d\times(0,\infty)^d\times\rsetp^{d\times d}\times\rsetp^d$, with
$\simplex_d$ defined by~(\ref{eq:simplex-def}).

In \cite{haas2004mixed}, it is shown that the equation satisfied by the
variance of a univariate NM($d$)-GARCH$(1,1)$ process admits a constant solution if and only if
\begin{equation} \label{eq:con:station:nmgarch}
\radius(\mathbf{A}+\mathbf{b}\boldsymbol{\gamma}^T)<1\eqsp,
\end{equation}
where, for any square matrix $\mathbf{M}$, $\radius(\mathbf{M})$ denotes the
spectral radius of $\mathbf{M}$. It follows that the existence of a weakly
stationary solution implies \eqref{eq:con:station:nmgarch} but it does not say
anything about the existence of stationary or weakly stationary solution. The
result below shows that \eqref{eq:con:station:nmgarch} is indeed a sufficient
condition for the existence of a stationary solution with finite variance. It
moreover provides with \autoref{thm:convergence-main} and
\autoref{rem:identifiability} the consistency of the MLE $\mlY{\mathbf{x}_1,n}$
for any $\mathbf{x}_1\in\Xset$.
%\subsubsection{Ergodicity}
\input{Halnmgarch-ergo}

%\subsubsection{Consistency}

\noindent\textbf{Step 2} We now show that Assumptions  %\ref{assum:momentCond},
  \ref{assum:continuity:Y:g-theta}, \ref{assum:continuity:Y:phi-theta} and \ref{assum:practical-cond:O} hold.

  Clearly, \ref{assum:continuity:Y:g-theta} and
  \ref{assum:continuity:Y:phi-theta} hold by definitions of
  $\psi^\theta_y(\mathbf{x})$ and $g^\theta(\mathbf{x};y)$. It remains to show
  ~\ref{assum:practical-cond:O}. Since $\Theta$ is compact, then
\begin{equation*}
\underline{\omega}\leq \min_{1\leq\ell\leq d}\boldsymbol{\omega}_\ell ,
\; |\boldsymbol{\omega}|\leq\overline{\omega}, \,
\underline{b} \leq |\mathbf{b}| \leq\bar b,\,
\radius(\mathbf{A}+\mathbf{b}\boldsymbol{\gamma}^T)\le\bar{\rho}, \;
\left\|\mathbf{A}+\mathbf{b}\boldsymbol{\gamma}^T\right\|\le L
\end{equation*}
for some $(\underline{\omega},\overline{\omega},\underline{b},\overline{b},\bar \rho) \in (0,\infty)^4\times(0,1)$ 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}
  \left\|\left(\mathbf{A}+\mathbf{b}\boldsymbol{\gamma}^T\right)^k\right\|\leq
  \bar C\,\bar\delta^k\;.
\end{equation}
We set
$\Xset_1=[\underline \omega,\infty)^d\subset\Xset$ so that
\ref{assum:practical-cond:O}\ref{assum:exist:practical-cond:subX}
holds. Moreover, for all
$(\theta,\mathbf{x},y)\in\Theta\times\Xset_1\times\Yset$,
$g^\theta(\mathbf{x};y)\le(2\pi\underline\omega)^{-1/2}$. Thus,
Condition~\ref{assum:practical-cond:O}\ref{assum:momentCond} holds.  Now let
$x_1\in\Xset$.
Using~\eqref{eq:psi:nm-garch:iterate},~(\ref{eq:uniform-spectral-raius})
and~(\ref{eq:norm-control2}), we have, for all $\mathbf{x}\in\Xset$, $y_{1:n}\in\Yset^n$ and $\theta\in\Theta$,
\begin{align*}
\left|\f{\chunk{y}1n}(\mathbf{x}_1)-\f{\chunk{y}1n}(\mathbf{x})\right|&= \left|\mathbf{A}^n(\mathbf{x}_1-\mathbf{x})\right|\\
&\leq \bar C\,\bar\delta^n\left|\mathbf{x}_1-\mathbf{x}\right|\eqsp.
  \end{align*}
Using that the norm defining $d$ is equivalent to the norm $|\cdot|$, we get~\ref{assum:practical-cond:O}\ref{assum:exist:practical-cond:O} with
$$
\bar\psi(\mathbf{x})= \bar C'\,\left|\mathbf{x}_1-\mathbf{x}\right|\;,
$$
for some positive constant $\bar
C'$. Hence~\ref{assum:practical-cond:O}\ref{item:psibar} holds and since
$$
\left|\mathbf{x}_1-\psi^\theta_y(\mathbf{x}_1)\right|\leq (L+1)\,|\mathbf{x}_1|+\overline{\omega}+y^2\bar b\;,
$$
we also get~\ref{assum:practical-cond:O}\ref{item:barphi:bar:phi} provided that
\begin{equation}
  \label{eq:phibar-nmgarch-temp}
\bar\phi(y)\geq (L+1)\,|\mathbf{x}_1|+\overline{\omega}+y^2\bar b \;.
\end{equation}
It is straightforward to show that,
% Moreover, for all $\theta\in\Theta$ and $y\in\Yset$,
% \begin{align*}
% \left\|\mathbf{x}_1-\psi^\theta_y(\mathbf{x}_1)\right\|_\star &\le \left\|\boldsymbol{\omega}\right\|_\star+\left(\lvert\lvert\lvert\mathbf{I}\rvert\rvert\rvert_\star+\lvert\lvert\lvert\mathbf{A}\rvert\rvert\rvert_\star\right)\left\|\mathbf{x}_1\right\|_\star+\left\|\mathbf{b}\right\|_\star y^2 \\
% &\le\bar\omega+\left(\lvert\lvert\lvert\mathbf{I}\rvert\rvert\rvert_\star+\bar\rho\right)\left\|\mathbf{x}_1\right\|_\star+\bar{b}y^2\eqsp.
% \end{align*}
% For all $\theta\in\Theta$, $\mathbf{x}\in\Xset_1$, $y\in\rset$, and $\ell\in\{1,\ldots,d\}$,
% \begin{align*}
% \frac{\partial\ln g^\theta}{\partial x_\ell}(\mathbf{x};y)&= \frac{\frac{\partial g^\theta}{\partial x_\ell}(\mathbf{x};y)}{g^\theta(\mathbf{x};y)}\\
% \frac{\partial g^\theta}{\partial x_\ell}(\mathbf{x};y)&=\frac{\partial} {\partial x_\ell}\left(\sum_{\ell=1}^{d}{\frac{\gamma_\ell}{(2\pi)^{1/2}}x_\ell^{-1/2}\exp\left(-\frac{y^2}{2x_\ell}\right)}\right)  \\
% &=\frac{\gamma_\ell}{2(2\pi)^{1/2}}(y^2-x_\ell)x_\ell^{-5/2}\exp\left(-\frac{y^2}{2x_\ell}\right)  \\
% \left|\frac{\partial g^\theta}{\partial x_\ell}(\mathbf{x};y)\right|&\le \frac{\gamma_\ell}{2(2\pi)^{1/2}}x_\ell^{-1/2}\exp\left(-\frac{y^2}{2x_\ell}\right)\left(\frac{y^2}{x_\ell^2}+\frac{1}{x_\ell}\right) \\
% &\le \frac{1}{2}g^\theta(\mathbf{x};y)\left(\frac{y^2}{\underline{\omega}^2}+\frac{1}{\underline{\omega}}\right).
% \end{align*}
% Hence
 for  all $\theta\in\Theta$, $\mathbf{x}\in\Xset_1$, $y\in\rset$, and $\ell\in\{1,\ldots,d\}$,
\begin{align*}
\left|\frac{\partial\ln g^\theta}{\partial x_\ell}(\mathbf{x};y)\right|&\le\frac{1}{2}\left(\frac{y^2}{\underline{\omega}^2}+\frac{1}{\underline{\omega}}\right)\eqsp.
\end{align*}
Thus, by the mean value theorem,  for all $\theta\in\Theta$, $(\mathbf{x},\mathbf{x}')\in\Xset_1\times\Xset_1$ and $y\in\Yset$,
\begin{align*}
\left|\ln g^\theta(\mathbf{x};y)-\ln g^\theta(\mathbf{x}';y)\right|&\le \frac{1}{2}\left(\frac{y^2}{\underline{\omega}^2}+\frac{1}{\underline{\omega}}\right)\,|\mathbf{x}-\mathbf{x}'|
\end{align*}
We thus obtain~\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} by setting $C=0$,
$$
H(u)=\sup_{d(\mathbf{x},\mathbf{x}')\leq u}|\mathbf{x}-\mathbf{x}'|\;,
$$
and
$$
\bar\phi(y)=(L+1)\,|\mathbf{x}_1|+\overline{\omega}+1/(2\underline{\omega})+y^2(\bar b+\underline{\omega}^2)\;.
$$
In addition, for all $\theta\in\Theta$ and $\mathbf{x}\in\Xset$, we have
$$
\int y^2 G^\theta(x,\rmd y) = \boldsymbol{\gamma}^T\mathbf{x} \;.
$$
Hence, using~(\ref{eq:Vbar-nmgarch}) with the above definitions, we
obtain~\ref{assum:practical-cond:O}\ref{item:22or23-barphi-V} and the proof is concluded.
\end{proof}




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