X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/16dcc.git/blobdiff_plain/3147e770da18dc984025737bc802ba0e0bff6977..c069ead2bda7032833d2d7f3d0851906ec4d22f0:/stopping.tex

diff --git a/stopping.tex b/stopping.tex
index aa13c9a..a6b821a 100644
--- a/stopping.tex
+++ b/stopping.tex
@@ -65,6 +65,7 @@ distribution induced by the $X$-th row of $P$. If the Markov chain induced by
 $P$ has a stationary distribution $\pi$, then we define
 $$d(t)=\max_{X\in\Bool^{\mathsf{N}}}\tv{P^t(X,\cdot)-\pi}.$$
 
+\ANNOT{incohÃ©rence de notation $X$ : entier ou dans $B^N$ ?}
 and
 
 $$t_{\rm mix}(\varepsilon)=\min\{t \mid d(t)\leq \varepsilon\}.$$
@@ -74,7 +75,7 @@ $$t_{\rm mix}(\varepsilon)=\min\{t \mid d(t)\leq \varepsilon\}.$$
 %% 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. 
 
 
@@ -117,7 +118,6 @@ $$\P_X(X_\tau=Y)=\pi(Y).$$
 
 \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 $\tau$. The following result will be useful~\cite[Proposition~6.10]{LevinPeresWilmer2006},
 
@@ -406,7 +406,7 @@ $\textit{fair}\leftarrow\emptyset$\;
 \end{algorithm}
 
 Practically speaking, for each number $\mathsf{N}$, $ 3 \le \mathsf{N} \le 16$, 
-10 functions have been generaed according to method presented in section~\ref{sec:hamilton}. For each of them, the calculus of the approximation of $E[\ts]$
+10 functions have been generated according to method presented in section~\ref{sec:hamilton}. For each of them, the calculus of the approximation of $E[\ts]$
 is executed 10000 times with a random seed. The Figure~\ref{fig:stopping:moy}
 summarizes these results. In this one, a circle represents the 
 approximation of $E[\ts]$ for a given $\mathsf{N}$.
