 \begin{theorem}\label{theo:ergo-convergence:nmgarch}
  Suppose that all $\theta=\left(\boldsymbol{\gamma}, \boldsymbol{\omega}, \mathbf{A}, \mathbf{b}\right)$ in $\Theta$ satisfy
  Condition~\eqref{eq:con:station:nmgarch}. 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 any norm on $\Xset$.
\end{theorem}


\begin{proof} In this proof section, we set
  \begin{equation}
    \label{eq:Vbar-nmgarch}
\bar V(\mathbf{x})=|\mathbf{x}|= \sum_{\ell=1}^d |x_\ell|\;,
  \end{equation}
for all $\mathbf{x}=(x_\ell)\in\Xset$. As
in~\autoref{theo:ergo-converge:nbigarch}, we divide the proof in two
steps.

\noindent\textbf{Step 1.} We first show that
Assumptions~\ref{assum:gen:identif:unique:pi} and \ref{ass:21-lyapunov} hold
with the above $\bar V$ by applying \autoref{thm:ergodicity}. Define $V$ on
$\Xset$ by setting 
$$
V(\mathbf{x})= (\mathbf{1}+\mathbf{x}_0)^T\mathbf{x} \;,
$$
where $\mathbf{1}$ is the vector of $\Xset$ with all entries equal to 1 and
$\mathbf{x}_0$ is defined by
$$
\mathbf{1}+\mathbf{x}_0 =
(\mathbf{I}-\left(\mathbf{A}+\mathbf{b}\boldsymbol{\gamma}^T\right)^T)^{-1}\mathbf{1} \;.
$$
We indeed note that by Condition~(\ref{eq:con:station:nmgarch}) the above
inversion is well defined and moreover
$$
(\mathbf{I}-(\mathbf{A}+\mathbf{b}\boldsymbol{\gamma}^T)^T)^{-1} = \mathbf{I}+ \sum_{k\geq1}
\left(\mathbf{A}^T+\boldsymbol{\gamma}\mathbf{b}^T\right)^k \;,
$$
and, since $\mathbf{A}$, $\mathbf{b}$, $\boldsymbol{\gamma}$ all have
non-negative entries, it follows that $\mathbf{x}_0$ has non-negative
entries. Thus, for all $\mathbf{x}=(x_\ell)\in\Xset$,
$$
\bar V(\mathbf{x}) = \mathbf{1}^T\mathbf{x} \leq V(\mathbf{x}) \;,
$$
so that $\bar V\lesssim V$. Hence by \autoref{thm:ergodicity}, we thus 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} with $V$ defined as above for a given
$\theta=\left(\boldsymbol{\gamma}, \boldsymbol{\omega}, \mathbf{A},
  \mathbf{b}\right)\in\Theta$ (so we drop $\theta$ in the notation in the
remaining of \textbf{Step 1}). Condition~\ref{ass-ergo-Xset} holds for any
distance $d$ associated to a norm on the finite dimensional space $\Xset$. (The precise choice of $d$ is
postponed to the verification of~\ref{assum:bound:rho}.) We have
\begin{align*}
RV(\mathbf{x})
&=\int V(\boldsymbol{\omega}+\mathbf{A}\mathbf{x}+y^2\mathbf{b})\;G(\mathbf{x},\rmd y)\\
&=
(\mathbf{1}+\mathbf{x}_0)^T\boldsymbol{\omega}+(\mathbf{1}+\mathbf{x}_0)^T\left(\mathbf{A}+\mathbf{b}\boldsymbol{\gamma}^T\right)\mathbf{x}\\
&=V(\boldsymbol{\omega})+\mathbf{1}^T(\mathbf{I}-\left(\mathbf{A}+\mathbf{b}\boldsymbol{\gamma}^T\right))^{-1}\left(\mathbf{A}+\mathbf{b}\boldsymbol{\gamma}^T-\mathbf{I}+\mathbf{I}\right)\mathbf{x}\\
&=V(\boldsymbol{\omega})+\mathbf{x}_0^T\mathbf{x} \\
&\leq V(\boldsymbol{\omega})+\lambda V(\mathbf{x})\;,
\end{align*}
where we set
$\lambda=\max_\ell\left\{\mathbf{x}_{0,\ell}/(1+\mathbf{x}_{0,\ell})\right\}<1$. Hence~\ref{assum:weakFeller-V}
holds. Condition \ref{assum:weakFeller}
easily follows from the continuity of the Gaussian distribution with respect to
its variance parameter. We now prove \ref{assum:reachable}.
From~\eqref{eq:notationItere:f:cl} and \eqref{eq:psi:nm-garch},
we have for all $n\geq1$, $y_{0:n-1}\in\Yset^{n}$ and $\mathbf{x}\in\Xset$,
\begin{equation}\label{eq:psi:nm-garch:iterate}
\f{\chunk{y}{0}{n-1}}(\mathbf{x})=
\mathbf{A}^{n}\mathbf{x}+\sum_{j=0}^{n-1}
\mathbf{A}^j (\boldsymbol{\omega}+y_{n-1-j}^2\mathbf{b})\eqsp.
\end{equation}
Let us use the norm
$$
\|\mathbf{M}\|= \max_j \sum_i |\mathbf{M}_{i,j}|=\sup_{|\mathbf{x}|\leq1}|\mathbf{M}\mathbf{x}|
$$ 
on $d\times d$ matrices. Note that
by~(\ref{eq:con:station:nmgarch}), there exists $\delta\in(0,1)$ and $c>0$ such
that, for any $k\geq1$,
\begin{equation}
  \label{eq:norm-control}
\left\|\left(\mathbf{A}+\mathbf{b}\boldsymbol{\gamma}^T\right)^k\right\|\leq c\;
\delta^k \;.
\end{equation}
Using that  $\mathbf{A}$, $\mathbf{b}$, $\boldsymbol{\gamma}$ all have
nonnegative entries, we have
\begin{equation}
  \label{eq:norm-control2}
\left\|\mathbf{A}^k\right\|\leq\left\|\left(\mathbf{A}+\mathbf{b}\boldsymbol{\gamma}^T\right)^k\right\|\;.
\end{equation}
Hence $(\mathbf{I}-\mathbf{A})^{-1}=\mathbf{I}+\sum_{k\ge1}\mathbf{A}^k$ is well defined and we set
$\mathbf{x}_\infty=\mathbf{(I-A)}^{-1}\boldsymbol{\omega}$ so that,
with~(\ref{eq:psi:nm-garch}), we have
$$
\f{\chunk{y}{0}{n-1}}(\mathbf{x})-\mathbf{x}_\infty=
\mathbf{A}^{n}\mathbf{x}+ \sum_{j\geq n}\mathbf{A}^j \boldsymbol{\omega} +
\sum_{j=0}^{n-1}
y_{n-1-j}^2\mathbf{A}^j \mathbf{b}\;.
$$
Then, using definition~(\ref{eq:def-Px}), we get that, $\bar\PP_{\mathbf{x}}$\as, for all $n\geq1$, 
\begin{align*}
  |\mathbf{X}_n-\mathbf{x}_\infty|&=\left|\f[]{\chunk{Y}0{n-1}}(\mathbf{x})-\mathbf{x}_\infty\right|\\
&\leq
\left|\mathbf{A}^n(\mathbf{x}-\mathbf{x}_\infty)\right|+ \sum_{j\geq n}\left|\mathbf{A}^j \boldsymbol{\omega}\right|+
\left(\max_{0\leq j\leq n-1}Y_j^2\right)\,\sum_{j=0}^{n-1}\left|\mathbf{A}^j\mathbf{b}\right|\;.
\end{align*}
With~(\ref{eq:norm-control}) and~(\ref{eq:norm-control2}), this implies
$$
\bar\PP_{\mathbf{x}}\left(|\mathbf{X}_n-\mathbf{x}_\infty|\leq c \left[\delta^n
    \left(|\mathbf{x}-\mathbf{x}_\infty|+\frac{|\boldsymbol{\omega}|}{1-\delta}\right)+\frac{|\mathbf{b}|}{1-\delta}\,\max_{0\leq
      j\leq n-1}Y_j^2\right]\right) = 1  \;.
$$
To obtain \ref{assum:reachable}, it is sufficient to observe that, since $g$
takes positive values in~(\ref{eq:g-theta:nm-garch}), for any
positive $\epsilon$, $\mathbf{x}\in\Xset$ and any $n\geq1$,
$$
\bar\PP_{\mathbf{x}}\left(\max_{0\leq j\leq n-1}Y_j^2<\epsilon\right)>0 \;.
$$
Next we prove~\ref{assum:bound:rho}. We have
$$
\psi_y(\mathbf{x})-\psi_y(\mathbf{x'})=\mathbf{A}(\mathbf{x}-\mathbf{x'}) \;.
$$
Since~(\ref{eq:norm-control}) and~(\ref{eq:norm-control2}) imply that
$\radius(\mathbf{A})<1$, there exists a vector norm which makes $\mathbf{A}$
strictly contracting. Choosing the metric $d$ on $\Xset$ as the distance
corresponding to this norm, we get~\ref{assum:bound:rho}. To
show~\ref{assum:38--40:alpha-phi}, we again rely
on~\autoref{lem:det:alpha:gen}. Let us set $\Cset=(0,\infty)$ and
$\Sset=\{1,\ldots, d\}$ and define the probability measure $\mu$ on $\Sset$ by
$\mu(\{s\})=\gamma_s$, for all $s\in\Sset$. For all $(t,y)\in\Cset\times\Yset$,
let $j(t)=\frac{1}{(2\pi t)^{1/2}}$ and
$h(t;y)=\exp\left(-y^2/{2t}\right)$. Obviously, $j$ is one-to-one,
Relation~(\ref{eq:def-g-j-h-mu}) holds and $h$ 
satisfies ~\ref{item:F1}. Hence, ~\autoref{lem:det:alpha:gen} implies that
$\alpha$ and $\phi$ defined respectively for all $\mathbf{x}=(x_1, \dots,
x_d), \ \mathbf{x'}=(x_1', \dots, x_d')\in\Xset$ by
\begin{equation*}\label{eq:alpha:x:x'}
\alpha(\mathbf{x,x'})=\min_{1\le\ell\le d}{\left\{\left(\frac{x_\ell\wedge x'_\ell}{x_\ell\vee x'_\ell}\right)^\frac{1}{2}\right\}}\in(0,1] \ \ \text{and} \ \ \phi\mathbf{(x,x')}=(x_1\wedge x'_1,\ldots,x_d\wedge x'_d)\eqsp,
\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}.  For $\mathbf{x}=(x_1,
\dots, x_d), \ \mathbf{x'}=(x_1', \dots, x_d')\in\Xset$, we have
\begin{align*}
1-\alpha(\mathbf{x,x'}) &=1-\min_{1\le\ell\le d}{\left\{\left(1-\frac{|x_\ell-x'_\ell|}{x_\ell\vee x'_\ell}\right)^\frac{1}{2}\right\}}\\
&\leq \max_{1\le\ell\le d}{\left\{\frac{|x_\ell-x'_\ell|}{x_\ell\vee
      x'_\ell}\right\}}\\
&\leq  \min_{1\le\ell\le d}(x_\ell^{-1}\wedge x_\ell^{\prime-1})\;|\mathbf{x}-\mathbf{x}'|\\
& \leq W(\mathbf{x},\mathbf{x}')\;d(\mathbf{x},\mathbf{x}') \;,
\end{align*}
where $d$ is the distance previously defined and $W$ is defined by
$W(\mathbf{x},\mathbf{x}')=1\vee \left(c_d \, \min_{1\le\ell\le
    d}(x_\ell^{-1}\wedge x_\ell^{\prime-1})\right)$ with $c_d>0$ is
conveniently chosen (such a constant exists since $d$ is the distance
associated to a norm and $\Xset$ has finite dimension). Then
\ref{assum:38--40:alpha-phi}\ref{assum:definition-gamma-x} and
\ref{assum:38--40:alpha-phi}\ref{assum:hyp:beta:gene} hold and, since for all
$y\in\Yset$ and $x\in\Xset$, $\psi_y(\mathbf{x})$ has all its entries bounded
from below by the positive entries of $\boldsymbol{\omega}$,
$W(\psi_y(\mathbf{x}),\psi_y(\mathbf{x}'))$ is uniformly bounded over
$(\mathbf{x, x'}, y)\in\Xset\times\Xset\times\Yset$
and~\ref{assum:38--40:alpha-phi}\ref{assum:driftCond:W:new} holds. This
completes \textbf{Step 1}.




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