From: Sylvain C-V <contasss@loria.fr>
Date: Mon, 27 Jun 2016 19:02:31 +0000 (+0200)
Subject: Fusion modifs
X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/16dcc.git/commitdiff_plain/c069ead2bda7032833d2d7f3d0851906ec4d22f0

Fusion modifs
---

c069ead2bda7032833d2d7f3d0851906ec4d22f0
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--- a/stopping.tex
+++ b/stopping.tex
@@@ -70,9 -69,13 +70,13 @@@ an
  
  $$t_{\rm mix}(\varepsilon)=\min\{t \mid d(t)\leq \varepsilon\}.$$
  
- Intuitively speaking, $t_{\rm mix}$ is a mixing time 
- \textit{i.e.}, is the time until the matrix $X$ \ANNOT{pas plutôt $P$ ?} of a Markov chain  
- is $\epsilon$-close to a stationary distribution.
+ %% Intuitively speaking, $t_{\rm mix}$ is a mixing time 
+ %% \textit{i.e.}, is the time until the matrix $X$ of a Markov chain  
+ %% is $\epsilon$-close to a stationary distribution.
+ 
+ Intutively speaking,  $t_{\rm mix}(\varepsilon)$ is the time/steps required
 -to be sure to be $\varepsilon$-close to the staionary distribution, wherever
++to be sure to be $\varepsilon$-close to the stationary distribution, wherever
+ the chain starts. 
  
  
  
@@@ -114,9 -117,9 +118,8 @@@ $$\P_X(X_\tau=Y)=\pi(Y).$
  
  \subsection{Upper bound of Stopping Time}\label{sub:stop:bound}
  
--
- A stopping time $\tau$ is a \emph{strong stationary time} if $X_{\tau}$ is
- independent of $\tau$. 
+ A stopping time $\tau$ is a {\emph strong stationary time} if $X_{\tau}$ is
+ independent of $\tau$. The following result will be useful~\cite[Proposition~6.10]{LevinPeresWilmer2006},
  
  
  \begin{thrm}\label{thm-sst}