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-\newcommand {\tv}[1] {\lVert #1 \rVert_{\rm TV}}
-\def \top {1.8}
-\def \topt {2.3}
-\def \P {\mathbb{P}}
-\def \ov {\overline}
-\def \ts {\tau_{\rm stop}}
-\begin{document}
-\label{firstpage}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Mathematical Backgroung}
-
-
-
-Let $\pi$, $\mu$ be two distribution on a same set $\Omega$. The total
-variation distance between $\pi$ and $\mu$ is denoted $\tv{\pi-\mu}$ and is
-defined by
-$$\tv{\pi-\mu}=\max_{A\subset \Omega} |\pi(A)-\mu(A)|.$$ It is known that
-$$\tv{\pi-\mu}=\frac{1}{2}\sum_{x\in\Omega}|\pi(x)-\mu(x)|.$$ Moreover, if
-$\nu$ is a distribution on $\Omega$, one has
-$$\tv{\pi-\mu}\leq \tv{\pi-\nu}+\tv{\nu-\mu}$$
-
-Let $P$ be the matrix of a markov chain on $\Omega$. $P(x,\cdot)$ is the
-distribution induced by the $x$-th row of $P$. If the markov chain induced by
-$P$ has a stationary distribution $\pi$, then we define
-$$d(t)=\max_{x\in\Omega}\tv{P^t(x,\cdot)-\pi},$$
-and
-
-$$t_{\rm mix}(\varepsilon)=\min\{t \mid d(t)\leq \varepsilon\}.$$
-One can prove that
-
-$$t_{\rm mix}(\varepsilon)\leq \lceil\log_2(\varepsilon^{-1})\rceil t_{\rm mix}(\frac{1}{4})$$
-
-It is known that $d(t+1)\leq d(t)$.
%% A {\it coupling} with transition matrix $P$ is a process $(X_t,Y_t)_{t\geq 0}$
%% with $\tau_{\rm couple}=\min_t\{X_t=Y_t\}$.
-Let $(X_t)_{t\in \mathbb{N}}$ be a sequence of $\Omega$ valued random
-variables. A $\mathbb{N}$-valued random variable $\tau$ is a {\it stopping
- time} for the sequence $(X_i)$ if for each $t$ there exists $B_t\subseteq
-\omega^{t+1}$ such that $\{tau=t\}=\{(X_0,X_1,\ldots,X_t)\in B_t\}$.
-
-Let $(X_t)_{t\in \mathbb{N}}$ be a markov chain and $f(X_{t-1},Z_t)$ a
-random mapping representation of the markov chain. A {\it randomized
- stopping time} for the markov chain is a stopping time for
-$(Z_t)_{t\in\mathbb{N}}$. It he markov chain is irreductible and has $\pi$
-as stationary distribution, then a {\it stationay time} $\tau$ is a
-randomized stopping time (possibily depending on the starting position $x$),
-such that the distribution of $X_\tau$ is $\pi$:
-$$\P_x(X_\tau=y)=\pi(y).$$
-
-\begin{proposition}
-If $\tau$ is a strong stationary time, then $d(t)\leq \max_{x\in\Omega}
-\P_x(\tau > t)$.
-\end{proposition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Random walk on the modified Hypercube}
The function $\ov{h}$ is said {\it square-free} if for every $x\in E$,
$\ov{h}(\ov{h}(x))\neq x$.
-\begin{lemma}\label{lm:h}
+\begin{Lemma}\label{lm:h}
If $\ov{h}$ is bijective and square-free, then $h(\ov{h}^{-1}(x))\neq h(x)$.
-\end{lemma}
+\end{Lemma}
-\begin{proof}
+\begin{Proof}
-\end{proof}
+\end{Proof}
Let $Z$ be a random variable over
$\{1,\ldots,N\}\times\{0,1\}$ uniformaly distributed. For $X\in \Omega$, we
the markov chain $(X_t)$.
-\begin{lemma}
+\begin{Lemma}
The integer $\ts$ is a strong stationnary time.
-\end{lemma}
+\end{Lemma}
\begin{proof}
Let $\tau_\ell$ be the first time that $\ell$ is fair. The random variable
lemma.
\end{proof}
-\begin{proposition} \label{prop:stop}
+\begin{Theo} \label{prop:stop}
If $\ov{h}$ is bijective and square-free, then
$E[\ts]\leq 8N^2+ N\ln (N+1)$.
-\end{proposition}
+\end{Theo}
For each $x\in \Omega$ and $\ell\in\{1,\ldots,N\}$, let $S_{x,\ell}$ be the
random variable counting the number of steps done until reaching from $x$ a state where
$$\lambda_h=\max_{x,\ell} S_{x,\ell}.$$
-\begin{lemma}\label{prop:lambda}
+\begin{Lemma}\label{prop:lambda}
If $\ov{h}$ is a square-free bijective function, then one has $E[\lambda_h]\leq 8N^2.$
-\end{lemma}
+\end{Lemma}
\begin{proof}
For evey $x$, every $\ell$, one has $\P(S_{x,\ell})\leq 2)\geq
Let $\ts^\prime$ be the first time that there are exactly $N-1$ fair
elements.
-\begin{lemma}\label{lm:stopprime}
+\begin{Lemma}\label{lm:stopprime}
One has $E[\ts^\prime]\leq N \ln (N+1).$
-\end{lemma}
+\end{Lemma}
\begin{proof}
This is a classical Coupon Collector's like problem. Let $W_i$ be the
$E[\ts^\prime]\leq N (-\frac{1}{2}+\ln(N+1))\leq N\ln(N+1)$.
\end{proof}
-One can now prove Proposition~\ref{prop:stop}.
+One can now prove Theo~\ref{prop:stop}.
\begin{proof}
One has $\ts\leq \ts^\prime+\lambda_h$. Therefore,
-Proposition~\ref{prop:stop} is a direct application of
+Theorem~\ref{prop:stop} is a direct application of
lemma~\ref{prop:lambda} and~\ref{lm:stopprime}.
\end{proof}
