X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/16dcc.git/blobdiff_plain/bd2919c6b5810121da30faafc9f3b8c6f0155e9e..31468b39be240f447015be22c252d515b2dcfdac:/stopping.tex

diff --git a/stopping.tex b/stopping.tex
index 989bb9e..7514341 100644
--- a/stopping.tex
+++ b/stopping.tex
@@ -65,12 +65,13 @@ 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{incohérence de notation $X$ : entier ou dans $B^N$ ?}
 and
 
 $$t_{\rm mix}(\varepsilon)=\min\{t \mid d(t)\leq \varepsilon\}.$$
 
 Intuitively speaking, $t_{\rm mix}$ is a mixing time 
-\textit{i.e.}, is the time until the matrix $X$ of a Markov chain  
+\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.
 
 
@@ -114,7 +115,7 @@ $$\P_X(X_\tau=Y)=\pi(Y).$$
 \subsection{Upper bound of Stopping Time}\label{sub:stop:bound}
 
 
-A stopping time $\tau$ is a {\emph strong stationary time} if $X_{\tau}$ is
+A stopping time $\tau$ is a \emph{strong stationary time} if $X_{\tau}$ is
 independent of $\tau$. 
 
 
@@ -401,7 +402,7 @@ $\textit{fair}\leftarrow\emptyset$\;
 \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 $\mathsf{N}$.