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\startlocaldefs
\input{definition}
\externaldocument{fmpv_clt}
\endlocaldefs

\theoremstyle{plain}
\newtheorem*{unlemma}{Lemma}
\newtheorem*{unproposition}{Proposition}
\newtheorem*{untheorem}{Theorem}


\begin{document}
\begin{frontmatter}
% "Title of the paper"
\title{Supplement to paper "A Central Limit Theorem for Adaptive and Interacting Markov Chains''}
\runtitle{Supplement}
\begin{aug}
 \author{\fnms{G.}
    \snm{Fort}\thanksref{t1,T1,m1}\ead[label=e1]{gersende.fort@telecom-paristech.fr}},
  \author{\fnms{E.}
    \snm{Moulines}\thanksref{T1,m2}\ead[label=e2]{eric.moulines@telecom-paristech.fr}},
  \author{\fnms{P.}  \snm{Priouret}\thanksref{m3}
    \ead[label=e3]{priouret@ccr.jussieu.fr}} \and
    \author{\fnms{P.} \snm{Vandekerkhove} \ead[label=e4]{pierre.vandek@univ-mlv.fr} \thanksref{m4} }



  \thankstext{t1}{Corresponding author}
  \thankstext{T1}{This
    work is partially supported by the French National Research Agency, under
    the program ANR-08-BLAN-0218 BigMC}
  \runauthor{G. Fort et al.}

  \affiliation{CNRS \& TELECOM ParisTech \thanksmark{m1}\thanksmark{m2}, Univ.
    Pierre et Marie Curie\thanksmark{m3}, Univ. Paris Est\thanksmark{m4}}
  

  \address{LTCI, TELECOM ParisTech \& CNRS, \\
    46 rue Barrault, \\
    75634 Paris Cedex 13, \\
    France \\ \printead{e1} \\ \printead{e2}}
  \address{LPMA, Univ. Pierre et Marie Curie, \\  Boîte courrier 188 \\ 75252 PARIS Cedex 05, France,\\
    \printead{e3}} \address{LAMA, Univ. Paris Est \\ 5 Bd Descartes
    Champs-sur Marne \\
    77454 Marne-la-Vallée Cedex 2, France.}
\end{aug}
\end{frontmatter}


\section{The variance of completely degenerated $V$-statistics of asymptotically stationary Markov chains}
The purpose of this section is to provide a bound for the variance of a 
completely degenerated V-statistics of an ergodic Markov Chain.

\section*{Notations and definitions}
For $\ell$ an integer and $\sequencen{y_n}$ a sequence, set $\chunk{y}{1}{\ell}\eqdef (y_{1},\dots,y_\ell)$. 
Let $(\Yset,\Ysigma)$ be a  measurable space.
For any integer $p$ and any $\mu_1, \dots, \mu_p$,
$p$ (signed) finite measures on $(\Yset,\Ysigma)$, denote by $\mu_1 \otimes
\dots \otimes \mu_p \eqdef \bigotimes_{i=1}^p \mu_i$, the product measure on
$(\Yset^p, \Ysigma^{\otimes p})$.  For $\mu$ a (signed) finite measure on
$(\Yset,\Ysigma)$, define $\mu^{\otimes p} \eqdef \mu \otimes \dots \otimes
\mu$.
Let $\{Y_{n}\}_{n=0}^{\infty}$ be a sequence of random variables with values in $(\Yset,\Ysigma)$.
We denote by $\pi_n$ the empirical measure associated to $\chunk{Y}{1}{n}$,
\[
\pi_n \eqdef n^{-1} \sum_{i=1}^n \delta_{Y_i} \eqsp,
\]
where for any $y \in \Yset$, $\delta_y$ denotes the Dirac mass at $y \in \Yset$. 

\paragraph{$V$-statistics} Let $h: \Yset^m \to \Rset$ be a measurable function and $\pi$ be a distribution on $(\Yset,\Ysigma)$. 
Consider the following $V$-statistics, for $n \geq m$,
\begin{align*}
  \bar V_{n,m}[h] & \eqdef  \idotsint h(y_1,\dots,y_m) \bigotimes_{i=1}^m \left[ \pi_n(\rmd y_i) - \pi(\rmd y_i) \right] \\
  &= V_{n,m}[\Phi] \eqsp,
\end{align*}
where
\begin{align}
\label{eq:definition-Phi}
\Phi(\chunk{y}{1}{m}) & \eqdef \bigotimes_{i=1}^m \left( \delta_{y_i} - \pi \right)[h] \eqsp, \\
 V_{n,m}[\Phi] & \eqdef n^{-m} \sum_{1 \leq i_1,\dots, i_m \leq n}
\Phi(Y_{\chunk{i}{1}{m}}) \eqsp.
\end{align}
Note that by definition of $\bar V_{n,m}$, we can assume without loss of
generality that $h$ is symmetric in its arguments. Hereafter, $h$ is assumed to
be symmetric.  Then, the function $\Phi$ is symmetric and, for any
$(y_1,\dots,y_{m-1})$,
\[
\int \Phi(y_1,\dots,y_{m-1},y) \pi(\rmd y) = 0 \eqsp.
\]
A symmetric function satisfying this relation is said to be \emph{$\pi$-canonical}.




\paragraph{$U$-statistics} For $c \in \{1,\dots,m\}$, let us consider the $U$-statistics ($n\geq c$)
\begin{equation}
\label{eq:definition-U-stat}
U_{n,c}\left[\Phi_{m,c}\right] \eqdef  \binom{n}{c}^{-1} \sum_{1 \leq i_1 < \dots < i_c \leq n} \Phi_{m,c}(Y_{i_1},\dots,Y_{i_c}) \eqsp,
\end{equation}
with symmetric kernels
\begin{equation}
\label{eq:definition-Phimk}
\Phi_{m,c}(\chunk{y}{1}{c}) \eqdef \sum_{\stackrel{\nu_1+\dots+\nu_c=m}{\nu_i \geq 1}} \frac{m!}{\nu_1! \dots \nu_c!} \Phi(y^{(\nu_1)}_1,\dots,y^{(\nu_c)}_c) \eqsp,
\end{equation}
where for $j \in \{1,\dots,c\}$,
\begin{equation}
\label{eq:definition-y-j-nu-j}
y_j^{(\nu_j)} \eqdef  (\overbrace{y_j,\dots,y_j}^{\nu_j}) \eqsp.
\end{equation}



\paragraph{Hoeffding Decomposition}
Let $g: \Yset^m \to \Rset$ be a symmetric function such that $\pi^{{\otimes m}}(|g|) < \infty$. 
Define for any $c \in \{1,\dots,m-1\}$ the
measurable function $\pi_{c,m} g: \Yset^c \to \Rset$ given by
 \begin{equation}
 \label{eq:definition-pi-k-m}
 \pi_{c,m}g(y_{1}, \dots, y_{c}) \eqdef (\delta_{y_{1}}-\pi)\otimes \dots \otimes (\delta_{y_{c}}-\pi) \otimes \pi^{\otimes (m-c)}[g] \eqsp.
 \end{equation}
Set
 \begin{equation}
 \label{eq:definition-pi-m-m}
\pi_{0,m} g \eqdef \pi^{\otimes m}(g) \quad \text{and} \quad  \pi_{m,m} g(y_{1}, \dots, y_{m}) \eqdef \bigotimes_{i=1}^m (\delta_{y_{i}}-\pi)[g]
 \eqsp.
 \end{equation}
 Note that for any $c \in \{1,\dots,m\}$ $\pi_{c,m}h$ is a $\pi$-canonical
 function.
The Hoeffding decomposition amounts to decompose the U-statistics $U_{n,m}[g]$ 
\begin{equation}
\label{eq:decomposition-hoeffding-g}
U_{n,m}[g]= \sum_{c=0}^m \binom{m}{c} U_{n,c}\left[\pi_{c,m}g\right] \eqsp,
\end{equation}


\paragraph{Degeneracy} Observe that for $c \in \{1, \cdots, m-1 \}$, $\pi_{c,m} g$ is $\pi$-canonical {\it
  i.e,} for all $(y_1,\dots,y_{c-1})\in \Yset^{c-1}$, $y \mapsto \pi_{c,m}
g(y_1,\dots,y_{c-1},y)$ is $\pi$-integrable and
\begin{equation}
\label{eq:degenere}
\int \pi(\rmd y) \pi_{c,m} g(y_1,\dots,y_{c-1}, y)=0 \, .
\end{equation}
Let $d \in \{0,\dots,m\}$. We say that the function $g$ is $d$-degenerated if
$\pi_{0,m}g=\dots = \pi_{d-1,m} g \equiv 0$ and $ \pi_{d,m} g \not \equiv 0$.
By convention, we say that the function $g$ is $0$-degenerated if $\pi_{0,m}g \neq 0$
(\ie\ $\pi^{\otimes m} (g) \ne 0$).






\section*{Results}
For $\mu$ a probability distribution on $(\Yset,\Ysigma)$ and $P$ a Markov
transition kernel on $(\Yset,\Ysigma)$, denote by $\PP_\mu$ the distribution of
the Markov chain $\sequencen{Y_n}$ with initial distribution $\mu$ and
transition kernel $P$; let $\PE_{\mu}$ be the associated expectation. For $p>0$
and $Z$ a random variable measurable \wrt\ the $\sigma$-algebra $\sigma\left(
  \sequencen{Y_n} \right)$, set $\lpnorm{Z}{\mu}{p}\eqdef \left(\PE_{\mu}\left[
    |Z|^p \right]\right)^{1/p}$. 


The supremum norm of a function $f$ is denoted by $\supnorm{f}$. For any
integer $m$ define by $\mathcal{L}_\infty(\Yset^m)$  the set of bounded
measurable functions on $\Yset^m$.

For two probability measures $\mu_1,\mu_2$ on $(\Yset,\Ysigma)$, $\tvnorm{\mu_1
  -\mu_2}$ denotes the total variation distance.

We assume that the transition kernel $P$ satisfies the following assumption:
\begin{hypU}
\label{hyp:assumption-kernel}
The kernel $P$ is positive Harris recurrent and has a unique stationary
distribution $\pi$. In addition, there exist a measurable function $V: \Yset
\to \coint{1,+\infty}$ and a constant $\rho \in (0,1)$ such that for any probability
distributions $\mu$ and $\mu'$ on $(\Yset,\Ysigma)$  and any integer $k$,
\begin{equation}
\label{eq:contraction-V-norm}
\tvnorm{\mu P^k - \mu' P^k} \leq \rho^k \left[ \mu(V) + \mu'(V) \right] \eqsp,
\end{equation}
and
\begin{equation}
\label{eq:finite-moment-pi}
\pi(V) < \infty \eqsp.
\end{equation}
\end{hypU}
For $\mu$ a probability measure on $(\Yset,\Ysigma)$, set
\begin{equation}
\label{eq:definition-M}
M(\mu,V) \eqdef \sup_{k \geq 0} \mu P^k (V) \eqsp.
\end{equation}
Note that under U\ref{hyp:assumption-kernel}, $\pi(V) \leq M(\mu,V)$.
\begin{theo}\label{theo:bounded-case}
  Assume U\ref{hyp:assumption-kernel}.  Let $\mu$ be a probability measure on
  $(\Yset, \Ysigma)$ such that $M(\mu, V) < \infty$. Then there exists a
  sequence $\sequence{A}{m}$ of positive constants such that for any $n \geq 2
  m$,
\[
\sup_{h \in \mathcal{L}_\infty(\Yset^m), \supnorm{h}\leq 1 }\lpnorm{\bar V_{n,m}[h]}{\mu}{2} \leq
A_m \, n^{-m/2} \eqsp,
\]
where $\limsup_m \ln A_m / (m \ln m) < \infty$.
\end{theo}
\begin{proof}
  Let $h: \Yset^m \to \Rset$ be a bounded symmetric measurable function.  For $n
\geq m$, we may decompose the $V$-statistic as a sum of $U$-statistics (see
\cite[p.  41]{koroljuk:borovskich:1994})
\begin{equation}
\label{eq:V-statistic-linear-combination-U-stat}
\bar V_{n,m}[h]= \sum_{k=1}^m \binom{n}{k} n^{-m} U_{n,m}\left[\Phi_{m,k}\right] \eqsp,
\end{equation}
where the functions $\Phi_{m,k}$, $k \in \{1,\dots,m\}$ are defined in \eqref{eq:definition-Phimk}.
Using the Minkovski inequality, we therefore obtain the bound
\begin{multline}
\label{eq:borne-elementaire-V-stat}
\lpnorm{\bar V_{n,m}[h]}{\mu}{2} \leq \sum_{k=1}^m \binom{n}{k} n^{-m} \lpnorm{U_{n,m}[\Phi_{m,k}]}{\mu}{2} 
\\
\leq \sum_{k=1}^m \frac{n^{k-m}}{k!}  \lpnorm{U_{n,m}[\Phi_{m,k}]}{\mu}{2} 
\eqsp.
\end{multline}
For any $k \in \{0,\dots,m\}$ the degeneracy of the function $\Phi_{m,k}$
defined in \eqref{eq:definition-Phimk} is larger than
\[
\inf\left\{ \sum_{i=1}^k \1_{\{\nu_i =1\}}: \nu_j \geq 1, \nu_1 + \cdots + \nu_k = m
\right\} \eqsp;
\]
therefore,
\begin{equation*}
\label{eq:definition-dmk}
d_{m,k} \geq (2k - m) \vee 0 \eqsp.
\end{equation*}
%% In particular, when $k \leq m/2$, the degeneracy of $\Phi_{m,k}$ is zero (which means that $\pi^{\otimes m}[\Phi_{m,k}] \ne 0$.
According to~\cite[Corollary~2.2.]{fort:moulines:priouret:vdk:2011-spl},
\begin{equation}
\label{eq:second-order-moment-Phimk}
\lpnorm{U_{n,m}\left[ \Phi_{m,k} \right]}{\mu}{2} \leq \left| \pi^{\otimes m}(\Phi_{m,k}) \right| +  \supnorm{\Phi_{m,k}} C_{m,k} \ n^{- (d_{m,k} \vee 1)/2}\eqsp,
\end{equation}
where the constant $C_{m,k}$ is given by
\begin{equation*}
\label{eq:definition-Cmk}
C_{m,k} \eqdef \sqrt{M(\mu,V)} \sum_{c= d_{m,k} \vee 1}^m \binom{m}{c}  2^{5c/2+1} \sqrt{(2c)!} \, c! \, \left( \sum_{j \geq 0} (j+1)^c \rho^j \right)^{1/2}   \eqsp;
\end{equation*}
(we use that $n \geq 2c$ to obtain this upper bound).  Therefore, since
$\pi^{\otimes m}(\Phi_{m,k}) =0$ when $d_{m,k} >0$ and $\left| \pi^{\otimes
    m}(\Phi_{m,k}) \right| \leq \supnorm{\Phi_{m,k}}$ otherwise,
Eqs~(\ref{eq:borne-elementaire-V-stat}) and
(\ref{eq:second-order-moment-Phimk}) imply
\begin{equation*}
\lpnorm{\bar V_{n,m}[h]}{\mu}{2} \leq n^{-m/2} \left\{ \sum_{1 \leq k \leq m/2}
  \supnorm{\Phi_{m,k}} \, \frac{n^{-m/2+k}}{k!} + \sum_{1 \leq  k \leq m}
  \supnorm{\Phi_{m,k}} \frac{C_{m,k}}{k!}  \right\} \eqsp.
\end{equation*}
Observe that using the multinomial formula
\[
\sum_{\stackrel{\nu_1+\dots+\nu_k=m}{\nu_i \geq 1}} \frac{m!}{\nu_1! \dots
  \nu_k!}  \leq k^m \eqsp,
\]
it holds
\[
\supnorm{\Phi_{m,k}} \leq k^m \ \supnorm{\Phi} \leq (2 k)^m \supnorm{h}
\eqsp.
\]
In order to upper bound the constant $C_{m,k}$, observe that by the Stirling formula,
$\sqrt{(2m)!} m!  \asymp 2^{m+1} \pi^{3/4} m^{2m + 3/4} \exp(-2m)$ as $m \to
\infty$. Then, there exists a finite constant $\kappa$ such that $ \sqrt{(2m)!}
m! \leq \kappa 2^{1+m} \ m^{3/4+2m}$. In addition, for all $m \in \Nset$,
$$
\sum_{j \geq 0} (j+1)^m \rho^j \leq \frac{1}{\rho \, (-\ln(\rho))^{m+1} }
\frac{m ^{m+1}-(-\ln(\rho))^{m+1}}{m +\ln (\rho)} \eqsp.
$$
The proof now follows.
\end{proof}
We now consider the case when $h$ is not bounded. Define for any $q \geq 1$,
\[
B_q(h) \eqdef \sup_{(y_1, \cdots, y_m) \in \Yset^m} \frac{|h(y_1, \cdots,
  y_m)|}{\sum_{j=1}^m V^{1/q}(y_j)} \eqsp.
\]
We first establish an upper bound for the variance of $U$-statistics when the
kernel $h$ is a symmetric $d(h)$-degenerated function.
\begin{prop}
\label{prop:Ustat-degenerated}
  Assume U\ref{hyp:assumption-kernel}.  Let $\mu$ be a probability measure on
  $(\Yset, \Ysigma)$ such that $M(\mu, V) < \infty$. For any measurable
  symmetric and $d(h)$-degenerated function $h : \Yset^m \to \Rset$ such that
  $B_{1/\alpha}(h) < +\infty$ for some $\alpha \in (0,1/2]$,
\[
\lpnorm{U_{n,m}[h] - \pi^{\otimes m}(h)}{\mu}{2} \leq B_{1/\alpha}(h) \ \left(1
  + m \ \pi(V^\alpha) \right) \ \sum_{c=d(h) \vee 1}^m 2^c \ \binom{m}{c} \bar
C_{\alpha, c} \ \frac{n^{c/2}}{\binom{n}{c}} \eqsp,
\]
where
\begin{multline*}
  \bar C_{\alpha, c} \eqdef 2^{c/2+1-\alpha} \  c \ \sqrt{(2c)!}  \cdots \\
  \times \left( \sum_{k \geq 0} (k+1)^c \rho^{k(1-2\alpha)} \right)^{1/2}
  \left(p^{2\alpha} + p^{-(1-2\alpha)} \right)^{1/2} \sqrt{M(\mu,V)} \eqsp,
\end{multline*}
and $ p \eqdef 1/(2\alpha)-1$.
\end{prop}
\begin{proof}
  By the Hoeffding decomposition,
\[
U_{n,m}[h] - \pi^{\otimes m}(h) = \sum_{c=d(h) \vee 1}^m \binom{m}{c}
U_{n,c}[\pi_{c,m}h] \eqsp.
\]
For $c \geq 1$, $\pi_{c,m}h$ is symmetric and $\pi$-canonical. Therefore, by
\cite[Corollary 2.3]{fort:moulines:priouret:vdk:2011-spl},
\[
\lpnorm{U_{n,c}[\pi_{c,m}h]}{\mu}{2} \leq \bar C_{\alpha, c} \ 
\frac{n^{c/2}}{\binom{n}{c}} \ B_{1/\alpha}(\pi_{c,m}h) \eqsp.
\]
Finally, it holds
\[
B_{1/\alpha}(\pi_{c,m}h) \leq B_{1/\alpha}(h) \ 2^c \ \left(1 + m \ \pi(V^\alpha)
\right) \eqsp.
\]
This concludes the proof.
\end{proof}
For any integer $m$ and any $\alpha \in (0,1/2]$, let 
$\mathcal{L}_\alpha(\Yset^m)$ be the set of measurable functions on $\Yset^m$ such
that $B_{1/\alpha}(h) < \infty$.
\begin{theo}
\label{theo:unbounded-case}
Assume U\ref{hyp:assumption-kernel}.  Let $\mu$ be a probability measure on
$(\Yset, \Ysigma)$ such that $M(\mu, V) < \infty$.  Let $\alpha \in (0,1/2]$.
There exists a sequence $\sequence{A}{m}$ of positive constants such that for
any $n \geq  m$,
\[
\sup_{h \in \mathcal{L}_\alpha(\Yset^m), B_{1/\alpha}(h) \leq 1 }\lpnorm{\bar V_{n,m}[h]}{\mu}{2} \leq
A_m \, n^{-m/2} \eqsp,
\]
where $\limsup_m \ln A_m / (m \ln m) < \infty$.
\end{theo}
The proof of Theorem~\ref{theo:unbounded-case} is on the same lines as the
proof of Theorem~\ref{theo:bounded-case}. A bound similar to
(\ref{eq:second-order-moment-Phimk}) is obtained by using
Proposition~\ref{prop:Ustat-degenerated}. Details are omitted.
\section{Weak law of large number for adaptive and interacting Markov chains}
\newcounter{saveenum}
\begin{untheorem}[Theorem~\ref{theo:WeakLLN}]
%%\label{theo:WeakLLN}
Assume \textbf{A\ref{hyp:DefinitionProc:X:Theta}}, \textbf{A\ref{hyp:geometric-ergodicity}} and
let $\gamma \in (0,1)$. Assume that
\begin{enumerate}[(i)]
\item \label{theo:WeakLLN:item1} $\limsup_{n \to \infty} L_{\t_n} < \infty$, $\PP$-\as\ where $L_\t$ is
  defined in Lemma~\ref{lem:BoundCandRho} applied with the closed interval
  $[\gamma, 1]$.
\item \label{theo:WeakLLN:item2} $\limsup_{n \to \infty} \pi_{\t_n}(V^\gamma) < \infty$, $\PP$-\as\ .
\item \label{theo:WeakLLN:item3} $\sup_{k \geq 1}  \PE\left[V(X_k)\right] < \infty$.
\item \label{theo:WeakLLN:item4} $n^{-1} \sum_{k=1}^n  D_{V^{\gamma}}(\t_k, \t_{k-1}) V^{\gamma}(X_k) \plim 0$.
\setcounter{saveenum}{\value{enumi}}
\end{enumerate}
Let $F :\Xset \times \Theta \to \Rset$ be a measurable function s.t.
\begin{enumerate}[(i)]
\setcounter{enumi}{\value{saveenum}}
\item \label{theo:WeakLLN:item5} $\limsup_{n \to \infty} \fnorm{F_{\t_n}}{V^{\gamma}} < +\infty$.
\item \label{theo:WeakLLN:item6} $n^{-1} \sum_{k=1}^{n-1} \fnorm{F_{\t_k} - F_{\t_{k-1}}}{V^{\gamma}} V^{\gamma}(X_k) \plim 0$.
\end{enumerate}
Then,
\[
\frac{1}{n} \sum_{k=0}^{n-1} F_{\t_k}(X_k) - \frac{1}{n} \sum_{k=0}^{n-1} \int
\pi_{\t_k}(\rmd x) F_{\t_k}(x)  \plim 0 \eqsp.
\]
\end{untheorem}

 \begin{proof}
    We write $ \frac{1}{n} \sum_{k=0}^{n-1} F_{\t_k}(X_k)
   -  \frac{1}{n} \sum_{k=0}^{n-1} \int
 \pi_{\t_k}(\rmd x) F_{\t_k}(x)= \sum_{i=1}^3 T_{i,n}$ with
 \begin{align*}
   T_{1,n} &\eqdef \frac{1}{n}F(X_0, \t_0) - \frac{1}{n} \int
   \pi_{\t_{n-1}}(\rmd x) F_{\t_{n-1}}(x) \\
   T_{2,n} &\eqdef \frac{1}{n} \sum_{k=1}^{n-1} \{F_{\t_k}(X_k) -F_{\t_{k-1}}(X_k) \} \\
   T_{3,n} &\eqdef \frac{1}{n}\sum_{k=1}^{n-1} \left\{F_{\t_{k-1}}(X_k) - \int
     \pi_{\t_{k-1}}(\rmd x) F_{\t_{k-1}}(x) \right\} \eqsp.
 \end{align*}
 Consider first $T_{1,n}$.  $|F_{\t_0}(X_0)| < +\infty$ $\PP$-\as; in addition,
 by \eqref{theo:WeakLLN:item2} and \eqref{theo:WeakLLN:item5},
 \[
 \limsup_n |\int \pi_{\t_{n-1}}(\rmd x) F_{\t_{n-1}}(x)| < +\infty \eqsp,
 \]
 $\PP$-\as.  Then, $\lim_{n \to \infty} T_{1,n} =0 $ $\PP$-\as\

 Under the conditions \eqref{theo:WeakLLN:item5} and \eqref{theo:WeakLLN:item6},
 $T_{2,n}$ converges to zero in probability.

 Consider finally $T_{3,n}$: by (\ref{eq:PoissonEquation}), it holds
 \[
 \frac{1}{n} \sum_{k=1}^{n-1} \bigg \{ F_{\t_{k-1}}(X_k) - \int
 \pi_{\theta_{k-1}}(\rmd x) F_{\t_{k-1}}(x) \bigg \} = M_n + R_n + \tilde R_n
 \]
  and
 \begin{align*}
   M_n & \eqdef \frac{1}{n} \sum_{k=1}^{n-1} \{ \operpoisson_{\theta_{k-1}}
   F_{\t_{k-1}}(X_k) - P_{\theta_{k-1}} \operpoisson_{\theta_{k-1}}
   F_{\t_{k-1}}(X_{k-1})\}  \eqsp,\\
   R_n & \eqdef \frac{1}{n} \sum_{k=1}^{n-1} \{ P_{\theta_{k}}
   \operpoisson_{\theta_{k}} F_{\t_{k}}(X_{k}) -P_{\theta_{k-1}}
   \operpoisson_{\theta_{k-1}}
   F_{\t_{k-1}}(X_{k}) \}  \eqsp, \\
   \tilde R_n & \eqdef \frac{1}{n} P_{\theta_0} \operpoisson_{\theta_{0}}
   F_{\t_{0}}(X_0) - \frac{1}{n} P_{\theta_{n-1}} \operpoisson_{\theta_{n-1}}
   F_{\t_{n-1}}(X_{n-1})\eqsp.
 \end{align*}
 By construction, $\{ \operpoisson_{\theta_{k-1}} F_{\t_{k-1}}(X_k) -
 P_{\theta_{k-1}}\operpoisson_{\theta_{k-1}} F_{\t_{k-1}}(X_{k-1}), k \geq 1 \}$
 is a martingale-increment sequence.  Therefore, by \cite[Theorem
 2.18]{hall:heyde:1980}, $M_n \aslim 0$ provided that w.p.1.
 \begin{equation}
 \label{eq:theo:LGNgal:tool1}
 \sum_{k \geq 1} \frac{1}{k^{1/\gamma}} \CPE{\left| \operpoisson_{\theta_{k-1}} F_{\t_{k-1}}(X_k) -
 P_{\theta_{k-1}}\operpoisson_{\theta_{k-1}} F_{\t_{k-1}}(X_{k-1})   \right|^{1/\gamma}}{\F_{k-1}} < +\infty  \eqsp.
 \end{equation}
 Eq. (\ref{eq:Controle:FctPoisson}) and Jensen's inequality imply that ($1/\gamma
 >1$)
 \begin{align*}
   & \CPE{\left|\operpoisson_{\theta_{k-1}} F_{\t_{k-1}}(X_k) -
       P_{\theta_{k-1}}\operpoisson_{\theta_{k-1}} F_{\t_{k-1}}(X_{k-1})\right|^{1/\gamma}}{\F_{k-1}} \\
   & \quad \leq 2^{1/\gamma-1} \CPE{\left|\operpoisson_{\theta_{k-1}}
       F_{\t_{k-1}}(X_k) \right|^{1/\gamma} +
     \left|P_{\theta_{k-1}} \operpoisson_{\theta_{k-1}} F_{\t_{k-1}}(X_{k-1}) \right|^{1/\gamma}}{\F_{k-1}} \\
   & \quad \leq 2^{1/\gamma} \left( \fnorm{F_{\t_{k-1}} }{V^\gamma} \
     L_{\t_{k-1}}^2 \right)^{1/\gamma} \ P_{\t_{k-1}}V(X_{k-1})\eqsp.
 \end{align*}
 Under items \eqref{theo:WeakLLN:item1}, \eqref{theo:WeakLLN:item3} and
 \eqref{theo:WeakLLN:item5}, the series is finite $\PP$-\as and this concludes
 the proof of (\ref{eq:theo:LGNgal:tool1}).  Consider now the remainder term
 $R_n$.  By \cite[Lemma 4.2.]{fort:moulines:priouret:2010},
 \[
 \fnorm{P_\t \operpoisson_{\t} F_\t - P_{\t'} \operpoisson_{\t'}
   F_{\t'}}{V^\gamma} \leq 5 \left( L_\t \vee L_{\t'} \right)^6 \pi_\t(V^\gamma)
 D_{V^\gamma}(\t,\t') +  L_{\t'}^2 \ \fnorm{F_\t - F_{\t'}}{V^\gamma} \eqsp.
 \]
 Then,
 \begin{multline*}
   |R_n| \leq \frac{1}{n} \sum_{k=1}^{n-1} L_{\t_{k-1}}^2 \, \fnorm{F_{\t_k} -
     F_{\t_{k-1}}}{V^\gamma}V^\gamma(X_k) \\
   +\frac{5}{n} \sum_{k=1}^{n-1} \left( L_{\t_k} \vee L_{\t_{k-1}} \right)^6
   \pi_{\t_k}(V^\gamma) \ D_{V^\gamma}(\t_k, \t_{k-1}) V^\gamma(X_k) \
   \left(\fnorm{F_{\t_k}}{V^\gamma} \vee\fnorm{F_{\t_{k-1}}}{V^\gamma}
   \right) \eqsp.
 \end{multline*}
 Under the stated assumptions (all the items except
 item~\eqref{theo:WeakLLN:item3}), this term converges to zero in probability.
 Consider finally $\tilde R_n$. By~(\ref{eq:Controle:FctPoisson}),
 \begin{multline*}
   \frac{1}{n} \left| P_{\theta_0} \hF_{\theta_0}(X_0) - P_{\theta_{n-1}}
     \hF_{\theta_{n-1}}(X_{n-1}) \right| \\
   \leq \frac{\left( \fnorm{F_{\t_0}}{V^\gamma} \vee
       \fnorm{F_{\t_{n-1}}}{V^\gamma}\right)}{n} \ \bigg( L_{\t_0}^2 \ P_{\t_0}
   V^\gamma(X_0)+
   L_{\t_{n-1}}^2  \ P_{\t_{n-1}} V^\gamma(X_{n-1})\bigg)  \\
   \leq \frac{\left( \fnorm{F_{\t_0}}{V^\gamma} \vee
       \fnorm{F_{\t_{n-1}}}{V^\gamma}\right)}{n} \ \bigg( L_{\t_0}^2 \ P_{\t_0}
   V(X_0)+ L_{\t_{n-1}}^2 \ P_{\t_{n-1}} V(X_{n-1})\bigg) \eqsp.
 \end{multline*}
 By items~\eqref{theo:WeakLLN:item1}, \eqref{theo:WeakLLN:item3} and
 \eqref{theo:WeakLLN:item5}, $\tilde R_n$ converges to zero in probability.

 \end{proof}






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