X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rairo15.git/blobdiff_plain/bc87eea8f89e38db87cb1d4705c6cfbfa993e27c..3820ec6286eab552baad3b3d7c0451e1bbc42482:/preliminaries.tex

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@@ -94,57 +94,6 @@ P=\dfrac{1}{6} \left(
 \end{xpl}
 
 
-
-More generally, let $\pi$, $\mu$ be two distribution on $\Bool^n$. The total
-variation distance between $\pi$ and $\mu$ is denoted $\tv{\pi-\mu}$ and is
-defined by
-$$\tv{\pi-\mu}=\max_{A\subset \Bool^n} |\pi(A)-\mu(A)|.$$ It is known that
-$$\tv{\pi-\mu}=\frac{1}{2}\sum_{x\in\Bool^n}|\pi(x)-\mu(x)|.$$ Moreover, if
-$\nu$ is a distribution on $\Bool^n$, one has
-$$\tv{\pi-\mu}\leq \tv{\pi-\nu}+\tv{\nu-\mu}$$
-
-Let $P$ be the matrix of a Markov chain on $\Bool^n$. $P(x,\cdot)$ 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
-$$d(t)=\max_{x\in\Bool^n}\tv{P^t(x,\cdot)-\pi}.$$
-It is known that $d(t+1)\leq d(t)$. \JFC{references ? Cela a-t-il 
-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?}
-% $$t_{\rm mix}(\varepsilon)\leq \lceil\log_2(\varepsilon^{-1})\rceil t_{\rm mix}(\frac{1}{4})$$
-
-
-
-Let $(X_t)_{t\in \mathbb{N}}$ be a sequence of $\Bool^n$ 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 $\{\tau=t\}=\{(X_0,X_1,\ldots,X_t)\in B_t\}$. 
-In other words, the event $\{\tau = t \}$ only depends on the values of 
-$(X_0,X_1,\ldots,X_t)$, not on $X_k$ with $k > t$. 
- 
-
-\JFC{Je ne comprends pas la definition de randomized stopping time, Peut-on enrichir ?}
-
-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
-$(Z_t)_{t\in\mathbb{N}}$. If the Markov chain is irreducible and has $\pi$
-as stationary distribution, then a {\it stationary time} $\tau$ is a
-randomized stopping time (possibly depending on the starting position $x$),
-such that  the distribution of $X_\tau$ is $\pi$:
-$$\P_x(X_\tau=y)=\pi(y).$$
-
-
-\JFC{Ou ceci a-t-il ete prouvé}
-\begin{Theo}
-If $\tau$ is a strong stationary time, then $d(t)\leq \max_{x\in\Bool^n}
-\P_x(\tau > t)$.
-\end{Theo}
-
 % % Let us first recall the  \emph{Total Variation} distance $\tv{\pi-\mu}$,
 % % which is defined for two distributions $\pi$ and $\mu$ on the same set 
 % % $\Bool^n$  by: