$$
p(e) \left\{
\begin{array}{ll}
-= \frac{1}{3} \textrm{ if $e=(v,v)$ with $v \in \Bool^3$,}\\
-= \frac{1}{3} \textrm{ otherwise.}
+= \frac{2}{3} \textrm{ if $e=(v,v)$ with $v \in \Bool^3$,}\\
+= \frac{1}{6} \textrm{ otherwise.}
\end{array}
\right.
$$
The matrix $P$ of the Markov chain associated to the function $f^*$ and to its probability function $p$ is
\[
-P=\dfrac{1}{3} \left(
+P=\dfrac{1}{6} \left(
\begin{array}{llllllll}
-1&1&1&0&0&0&0&0 \\
-1&1&0&0&0&1&0&0 \\
-0&0&1&1&0&0&1&0 \\
-0&1&1&1&0&0&0&0 \\
-1&0&0&0&1&0&1&0 \\
-0&0&0&0&1&1&0&1 \\
-0&0&0&0&1&0&1&1 \\
-0&0&0&1&0&1&0&1
+4&1&1&0&0&0&0&0 \\
+1&4&0&0&0&1&0&0 \\
+0&0&4&1&0&0&1&0 \\
+0&1&1&4&0&0&0&0 \\
+1&0&0&0&4&0&1&0 \\
+0&0&0&0&1&4&0&1 \\
+0&0&0&0&1&0&4&1 \\
+0&0&0&1&0&1&0&4
\end{array}
\right)
\]
introduced. We further detail
a theoretical study on the length of the path
which is sufficient to follow to get a uniform distribution.
-
+Notice that for a general references on Markov chains
+see~\cite{LevinPeresWilmer2006}
+, and particularly Chapter~5 on stopping times.
and
$$t_{\rm mix}(\varepsilon)=\min\{t \mid d(t)\leq \varepsilon\}.$$
-% One can prove that
+One can prove that
-% $$t_{\rm mix}(\varepsilon)\leq \lceil\log_2(\varepsilon^{-1})\rceil t_{\rm mix}(\frac{1}{4})$$
+$$t_{\rm mix}(\varepsilon)\leq \lceil\log_2(\varepsilon^{-1})\rceil t_{\rm mix}(\frac{1}{4})$$
% It is known that $d(t+1)\leq d(t)$. \JFC{references ? Cela a-t-il
-% un intérêt dans la preuve ensuite.}
+% un intérêt dans la preuve ensuite.}
%and
% $$t_{\rm mix}(\varepsilon)=\min\{t \mid d(t)\leq \varepsilon\}.$$
-% One can prove that \JFc{Ou cela a-t-il été fait?}
+% One can prove that \JFc{Ou cela a-t-il été fait?}
% $$t_{\rm mix}(\varepsilon)\leq \lceil\log_2(\varepsilon^{-1})\rceil t_{\rm mix}(\frac{1}{4})$$
such that the distribution of $X_\tau$ is $\pi$:
$$\P_X(X_\tau=Y)=\pi(Y).$$
+A stopping time $\tau$ is a {\emph strong stationary time} if $X_{\tau}$ is
+independent of $\tau$.
+
-\begin{Theo}
+\begin{thrm}
If $\tau$ is a strong stationary time, then $d(t)\leq \max_{X\in\Bool^{\mathsf{N}}}
\P_X(\tau > t)$.
-\end{Theo}
+\end{thrm}
%Let $\Bool^n$ be the set of words of length $n$.
We define the Markov matrix $P_h$ for each line $X$ and
each column $Y$ as follows:
-$$\left\{
+\begin{equation}
+\left\{
\begin{array}{ll}
-P_h(X,X)=\frac{1}{{\mathsf{N}}} & \\
+P_h(X,X)=\frac{1}{2}+\frac{1}{2{\mathsf{N}}} & \\
P_h(X,Y)=0 & \textrm{if $(X,Y)\notin E_h$}\\
-P_h(X,Y)=\frac{1}{{\mathsf{N}}} & \textrm{if $X\neq Y$ and $(X,Y) \in E_h$}
+P_h(X,Y)=\frac{1}{2{\mathsf{N}}} & \textrm{if $X\neq Y$ and $(X,Y) \in E_h$}
\end{array}
\right.
-$$
+\label{eq:Markov:rairo}
+\end{equation}
We denote by $\ov{h} : \Bool^{\mathsf{N}} \rightarrow \Bool^{\mathsf{N}}$ the function
such that for any $X \in \Bool^{\mathsf{N}} $,
The function $\ov{h}$ is said {\it square-free} if for every $X\in \Bool^{\mathsf{N}}$,
$\ov{h}(\ov{h}(X))\neq X$.
-\begin{Lemma}\label{lm:h}
+\begin{lmm}\label{lm:h}
If $\ov{h}$ is bijective and square-free, then $h(\ov{h}^{-1}(X))\neq h(X)$.
-\end{Lemma}
+\end{lmm}
\begin{proof}
Let $\ov{h}$ be bijective.
\end{proof}
Let $Z$ be a random variable that is uniformly distributed over
-$\llbracket 1, {\mathsf{N}}$.
+$\llbracket 1, {\mathsf{N}} \rrbracket \times \Bool$.
For $X\in \Bool^{\mathsf{N}}$, we
-define, with $Z=i$,
+define, with $Z=(i,b)$,
$$
\left\{
\begin{array}{ll}
-%f(X,Z)=X\oplus (0^{{\mathsf{N}}-i}10^{i-1}) & \text{if } b=1 \text{ and } i\neq h(X),\\
-f(X,Z)=X\oplus (0^{{\mathsf{N}}-i}10^{i-1}) & \text{if $i\neq h(X)$},\\
+f(X,Z)=X\oplus (0^{{\mathsf{N}}-i}10^{i-1}) & \text{if } b=1 \text{ and } i\neq h(X),\\
f(X,Z)=X& \text{otherwise.}
\end{array}\right.
$$
-%%%%%%%%%%%%%%%%%%%%%%%%%%%ù
+%%%%%%%%%%%%%%%%%%%%%%%%%%%ù
%\section{Stopping time}
An integer $\ell\in \llbracket 1,{\mathsf{N}} \rrbracket$ is said {\it fair}
at time $t$ if there
-exists $0\leq j <t$ such that $Z_{j+1}=\ell$ and $h(X_j)\neq \ell$.
-In other words, there exist a date $j$ before $t$ where
-the random variable $Z$ is $l$
+exists $0\leq j <t$ such that $Z_{j+1}=(\ell,\cdot)$ and $h(X_j)\neq \ell$.
+In other words, there exist a date $j$ before $t$ where
+the first element of the random variable $Z$ is exactly $l$
(\textit{i.e.}, $l$ is the strategy at date $j$)
and where the configuration $X_j$ allows to traverse the edge $l$.
the Markov chain $(X_t)$.
-\begin{Lemma}
+\begin{lmm}
The integer $\ts$ is a strong stationary time.
-\end{Lemma}
+\end{lmm}
\begin{proof}
Let $\tau_\ell$ be the first time that $\ell$ is fair. The random variable
-$Z_{\tau_\ell}$ is of the form $\ell$ %with $\delta\in\{0,1\}$ and
-% such that
-% $b=1$ with probability $\frac{1}{2}$ and $b=0$ with probability
-% $\frac{1}{2}$.
-Since $h(X_{\tau_\ell-1})\neq\ell$ the value of the $\ell$-th
+$Z_{\tau_\ell}$ is of the form $(\ell,b)$ %with $\delta\in\{0,1\}$ and
+such that
+$b=1$ with probability $\frac{1}{2}$ and $b=0$ with probability
+$\frac{1}{2}$. Since $h(X_{\tau_\ell-1})\neq\ell$ the value of the $\ell$-th
bit of $X_{\tau_\ell}$
is $0$ or $1$ with the same probability ($\frac{1}{2}$).
$\ell$-th bit of $X_t$ is $0$ or $1$ with the same probability, proving the
lemma.\end{proof}
-\begin{Theo} \label{prop:stop}
+\begin{thrm} \label{prop:stop}
If $\ov{h}$ is bijective and square-free, then
-$E[\ts]\leq {\mathsf{N}}^2+ (\mathsf{N}+2)(\ln(\mathsf{N})+2)$.
-\end{Theo}
+$E[\ts]\leq 8{\mathsf{N}}^2+ 4{\mathsf{N}}\ln ({\mathsf{N}}+1)$.
+\end{thrm}
For each $X\in \Bool^{\mathsf{N}}$ and $\ell\in\llbracket 1,{\mathsf{N}}\rrbracket$,
let $S_{X,\ell}$ be the
random variable that counts the number of steps
from $X$ until we reach a configuration where
$\ell$ is fair. More formally
-$$S_{X,\ell}=\min \{t \geq 1\mid h(X_{t-1})\neq \ell\text{ and }Z_t=\ell \text{ and } X_0=X\}.$$
-
- We denote by
-$$\lambda_h=\max_{X,\ell} S_{X,\ell}.$$
+$$S_{X,\ell}=\min \{t \geq 1\mid h(X_{t-1})\neq \ell\text{ and }Z_t=(\ell,.)\text{ and } X_0=X\}.$$
+% We denote by
+% $$\lambda_h=\max_{X,\ell} S_{X,\ell}.$$
-\begin{Lemma}\label{prop:lambda}
-If $\ov{h}$ is a square-free bijective function, then the inequality
-$E[\lambda_h]\leq 2{\mathsf{N}}^2$ is established.
-\end{Lemma}
+\begin{lmm}\label{prop:lambda}
+Let $\ov{h}$ is a square-free bijective function. Then
+for all $X$ and
+all $\ell$,
+the inequality
+$E[S_{X,\ell}]\leq 8{\mathsf{N}}^2$ is established.
+\end{lmm}
\begin{proof}
-For every $X$, every $\ell$, one has $\P(S_{X,\ell}\leq 2)\geq
-\frac{1}{{\mathsf{N}}^2}$.
+For every $X$, every $\ell$, one has $\P(S_{X,\ell})\leq 2)\geq
+\frac{1}{4{\mathsf{N}}^2}$.
Let $X_0= X$.
Indeed,
\begin{itemize}
\item if $h(X)\neq \ell$, then
-$\P(S_{X,\ell}=1)=\frac{1}{{\mathsf{N}}}\geq \frac{1}{{\mathsf{N}}^2}$.
+$\P(S_{X,\ell}=1)=\frac{1}{2{\mathsf{N}}}\geq \frac{1}{4{\mathsf{N}}^2}$.
\item otherwise, $h(X)=\ell$, then
$\P(S_{X,\ell}=1)=0$.
But in this case, intuitively, it is possible to move
-from $X$ to $\ov{h}^{-1}(X)$ (with probability $\frac{1}{N}$). And in
+from $X$ to $\ov{h}^{-1}(X)$ (with probability $\frac{1}{2N}$). And in
$\ov{h}^{-1}(X)$ the $l$-th bit can be switched.
More formally,
since $\ov{h}$ is square-free,
$\ov{h}(X)=\ov{h}(\ov{h}(\ov{h}^{-1}(X)))\neq \ov{h}^{-1}(X)$. It follows
that $(X,\ov{h}^{-1}(X))\in E_h$. We thus have
-$P(X_1=\ov{h}^{-1}(X))=\frac{1}{{\mathsf{N}}}$. Now, by Lemma~\ref{lm:h},
+$P(X_1=\ov{h}^{-1}(X))=\frac{1}{2{\mathsf{N}}}$. Now, by Lemma~\ref{lm:h},
$h(\ov{h}^{-1}(X))\neq h(X)$. Therefore $\P(S_{x,\ell}=2\mid
-X_1=\ov{h}^{-1}(X))=\frac{1}{{\mathsf{N}}}$, proving that $\P(S_{x,\ell}\leq 2)\geq
-\frac{1}{{\mathsf{N}}^2}$.
+X_1=\ov{h}^{-1}(X))=\frac{1}{2{\mathsf{N}}}$, proving that $\P(S_{x,\ell}\leq 2)\geq
+\frac{1}{4{\mathsf{N}}^2}$.
\end{itemize}
-Therefore, $\P(S_{X,\ell}\geq 3)\leq 1-\frac{1}{{\mathsf{N}}^2}$. By induction, one
+Therefore, $\P(S_{X,\ell}\geq 3)\leq 1-\frac{1}{4{\mathsf{N}}^2}$. By induction, one
has, for every $i$, $\P(S_{X,\ell}\geq 2i)\leq
-\left(1-\frac{1}{{\mathsf{N}}^2}\right)^i$.
+\left(1-\frac{1}{4{\mathsf{N}}^2}\right)^i$.
Moreover,
since $S_{X,\ell}$ is positive, it is known~\cite[lemma 2.9]{proba}, that
$$E[S_{X,\ell}]=\sum_{i=1}^{+\infty}\P(S_{X,\ell}\geq i).$$
\P(S_{X,\ell}\geq 1)+\P(S_{X,\ell}\geq 2)+2 \sum_{i=1}^{+\infty}\P(S_{X,\ell}\geq 2i).$$
Consequently,
$$E[S_{X,\ell}]\leq 1+1+2
-\sum_{i=1}^{+\infty}\left(1-\frac{1}{{\mathsf{N}}^2}\right)^i=2+2({\mathsf{N}}^2-1)=2{\mathsf{N}}^2,$$
+\sum_{i=1}^{+\infty}\left(1-\frac{1}{4{\mathsf{N}}^2}\right)^i=2+2(4{\mathsf{N}}^2-1)=8{\mathsf{N}}^2,$$
which concludes the proof.
\end{proof}
-Let $\ts^\prime$ be the first time that there are exactly ${\mathsf{N}}-1$ fair
-elements.
+Let $\ts^\prime$ be the time used to get all the bits but one fair.
-\begin{Lemma}\label{lm:stopprime}
-One has $E[\ts^\prime]\leq (\mathsf{N}+2)(\ln(\mathsf{N})+2)$.
-\end{Lemma}
+\begin{lmm}\label{lm:stopprime}
+One has $E[\ts^\prime]\leq 4{\mathsf{N}} \ln ({\mathsf{N}}+1).$
+\end{lmm}
\begin{proof}
-This is a classical Coupon Collector's like problem. Let $W_i$
-be the time to obtain the $i$-th fair bit
-after $i-1$ fair bits have been obtained.
-One has $\ts^\prime=\sum_{i=1}^{{\mathsf{N}}}W_i$.
-
-At position $X$ with $i-1$ fair bits,
-we do not obtain a new fair if $Z$ is one of the $i-1$ already fair bits
-or if $Z$ is a new fair bit but $h(X)$ is $Z$.
-This occurs with probability
-$p
-= \frac{i-1}{{\mathsf{N}}} + \frac{n-i+1}{\mathsf{N}}.\frac{1}{\mathsf{N}}
-=\frac{i(\mathsf{N}-1) +1}{\mathsf{N^2}}
-$.
-The random variable $W_i$ has a geometric distribution
-\textit{i.e.}, $P(W_i = k) = p^{k-1}.(1-p)$ and
-$E(W_i) = \frac{\mathsf{N^2}}{i(\mathsf{N}-1) +1}$.
-Therefore
-$$E[\ts^\prime]=\sum_{i=1}^{{\mathsf{N}}}E[W_i]
-=\frac{\mathsf{N^2}}{\mathsf{N}(\mathsf{N}-1) +1} + \sum_{i=1}^{{\mathsf{N}}-1}E[W_i].$$
-
-A simple study of the function $\mathsf{N} \mapsto \frac{\mathsf{N^2}}{\mathsf{N}(\mathsf{N}-1) +1}$ shows that it is bounded by $\frac{4}{3} \leq 2$.
-For the second term, we successively have
-$$
-\sum_{i=1}^{{\mathsf{N}}-1}E[W_i]
-= \mathsf{N}^2\sum_{i=1}^{{\mathsf{N}}-1} \frac{1}{i(\mathsf{N}-1) +1}
-\leq \mathsf{N}^2\sum_{i=1}^{{\mathsf{N}}-1} \frac{1}{i(\mathsf{N}-1)}
-\leq \frac{\mathsf{N}^2}{\mathsf{N}-1}\sum_{i=1}^{{\mathsf{N}}-1} \frac{1}{i}
-\leq (\mathsf{N}+2)\sum_{i=1}^{{\mathsf{N}}-1} \frac{1}{i}
-$$
+This is a classical Coupon Collector's like problem. Let $W_i$ be the
+random variable counting the number of moves done in the Markov chain while
+we had exactly $i-1$ fair bits. One has $\ts^\prime=\sum_{i=1}^{{\mathsf{N}}-1}W_i$.
+ But when we are at position $X$ with $i-1$ fair bits, the probability of
+ obtaining a new fair bit is either $1-\frac{i-1}{{\mathsf{N}}}$ if $h(X)$ is fair,
+ or $1-\frac{i-2}{{\mathsf{N}}}$ if $h(X)$ is not fair.
+
+Therefore,
+$\P (W_i=k)\leq \left(\frac{i-1}{{\mathsf{N}}}\right)^{k-1} \frac{{\mathsf{N}}-i+2}{{\mathsf{N}}}.$
+Consequently, we have $\P(W_i\geq k)\leq \left(\frac{i-1}{{\mathsf{N}}}\right)^{k-1} \frac{{\mathsf{N}}-i+2}{{\mathsf{N}}-i+1}.$
+It follows that $E[W_i]=\sum_{k=1}^{+\infty} \P (W_i\geq k)\leq {\mathsf{N}} \frac{{\mathsf{N}}-i+2}{({\mathsf{N}}-i+1)^2}\leq \frac{4{\mathsf{N}}}{{\mathsf{N}}-i+2}$.
+
-It is well known that
-$\sum_{i=1}^{{\mathsf{N}}-1}\frac{1}{i}\leq 1+\ln({\mathsf{N}}-1)$.
-It follows that
-$2+(\mathsf{N}+2)\sum_{i=1}^{{\mathsf{N}}-1}\frac{1}{i}
-\leq
-2+(\mathsf{N}+2)(\ln(\mathsf{N}-1)+1)
-\leq
-(\mathsf{N}+2)(\ln(\mathsf{N})+2)$.
+It follows that
+$E[W_i]\leq \frac{4{\mathsf{N}}}{{\mathsf{N}}-i+2}$. Therefore
+$$E[\ts^\prime]=\sum_{i=1}^{{\mathsf{N}}-1}E[W_i]\leq
+4{\mathsf{N}}\sum_{i=1}^{{\mathsf{N}}-1} \frac{1}{{\mathsf{N}}-i+2}=
+4{\mathsf{N}}\sum_{i=3}^{{\mathsf{N}}+1}\frac{1}{i}.$$
+
+But $\sum_{i=1}^{{\mathsf{N}}+1}\frac{1}{i}\leq 1+\ln({\mathsf{N}}+1)$. It follows that
+$1+\frac{1}{2}+\sum_{i=3}^{{\mathsf{N}}+1}\frac{1}{i}\leq 1+\ln({\mathsf{N}}+1).$
+Consequently,
+$E[\ts^\prime]\leq
+4{\mathsf{N}} (-\frac{1}{2}+\ln({\mathsf{N}}+1))\leq
+4{\mathsf{N}}\ln({\mathsf{N}}+1)$.
\end{proof}
One can now prove Theorem~\ref{prop:stop}.
\begin{proof}
-One has $\ts\leq \ts^\prime+\lambda_h$. Therefore,
+Since $\ts^\prime$ is the time used to obtain $\mathsf{N}-1$ fair bits.
+Assume that the last unfair bit is $\ell$. One has
+$\ts=\ts^\prime+S_{X_\tau,\ell}$, and therefore
+$E[\ts] = E[\ts^\prime]+E[S_{X_\tau,\ell}]$. Therefore,
Theorem~\ref{prop:stop} is a direct application of
lemma~\ref{prop:lambda} and~\ref{lm:stopprime}.
\end{proof}
