Relecture Sylvain avec commentaires/questions

[16dcc.git] / stopping.tex

diff --git a/stopping.tex b/stopping.tex
index 989bb9eee1f7cb01b79d4b91bf71f0cc22f1e8f4..75143419beff3d5ac218fb587c8e322435ee45a7 100644 (file)
--- 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
 $$d(t)=\max_{X\in\Bool^{\mathsf{N}}}\tv{P^t(X,\cdot)-\pi}.$$
 
 $P$ 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 
 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.
 
 
 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}
 
 
 \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 $\tau$. 
 
 
 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$, 
 \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 $\mathsf{N}$.
 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 $\mathsf{N}$.