ajout du début des expériementations

[rairo15.git] / stopping.tex

diff --git a/stopping.tex b/stopping.tex
index e4136889a4471c8ddaa8fe68bb5245fa4878a646..d3b4f4d4c1ff44127c30b3dcd93765f54aa2e367 100644 (file)
--- a/stopping.tex
+++ b/stopping.tex
@@ -165,14 +165,16 @@ has been removed.
 
 We define the Markov matrix $P_h$ for each line $X$ and 
 each column $Y$  as follows:
 
 We define the Markov matrix $P_h$ for each line $X$ and 
 each column $Y$  as follows:

-$$\left\{
+\begin{equation}
+\left\{

 \begin{array}{ll}
 P_h(X,X)=\frac{1}{2}+\frac{1}{2{\mathsf{N}}} & \\
 P_h(X,Y)=0 & \textrm{if  $(X,Y)\notin E_h$}\\
 P_h(X,Y)=\frac{1}{2{\mathsf{N}}} & \textrm{if $X\neq Y$ and $(X,Y) \in E_h$}
 \end{array}
 \right.
 \begin{array}{ll}
 P_h(X,X)=\frac{1}{2}+\frac{1}{2{\mathsf{N}}} & \\
 P_h(X,Y)=0 & \textrm{if  $(X,Y)\notin E_h$}\\
 P_h(X,Y)=\frac{1}{2{\mathsf{N}}} & \textrm{if $X\neq Y$ and $(X,Y) \in E_h$}
 \end{array}
 \right.

-$$ 
+\label{eq:Markov:rairo}
+\end{equation} 

 
 We denote by $\ov{h} : \Bool^{\mathsf{N}} \rightarrow \Bool^{\mathsf{N}}$ the function 
 such that for any $X \in \Bool^{\mathsf{N}} $, 
 
 We denote by $\ov{h} : \Bool^{\mathsf{N}} \rightarrow \Bool^{\mathsf{N}}$ the function 
 such that for any $X \in \Bool^{\mathsf{N}} $, 


@@ -289,10 +291,10 @@ More formally,
 since $\ov{h}$ is square-free,
 $\ov{h}(X)=\ov{h}(\ov{h}(\ov{h}^{-1}(X)))\neq \ov{h}^{-1}(X)$. It follows
 that $(X,\ov{h}^{-1}(X))\in E_h$. We thus have
 since $\ov{h}$ is square-free,
 $\ov{h}(X)=\ov{h}(\ov{h}(\ov{h}^{-1}(X)))\neq \ov{h}^{-1}(X)$. It follows
 that $(X,\ov{h}^{-1}(X))\in E_h$. We thus have

-$P(X_1=\ov{h}^{-1}(X))=\frac{1}{2{\MATHSF{N}}}$. Now, by Lemma~\ref{lm:h},
+$P(X_1=\ov{h}^{-1}(X))=\frac{1}{2{\mathsf{N}}}$. Now, by Lemma~\ref{lm:h},

 $h(\ov{h}^{-1}(X))\neq h(X)$. Therefore $\P(S_{x,\ell}=2\mid
 $h(\ov{h}^{-1}(X))\neq h(X)$. Therefore $\P(S_{x,\ell}=2\mid

-X_1=\ov{h}^{-1}(X))=\frac{1}{2{\MATHSF{N}}}$, proving that $\P(S_{x,\ell}\leq 2)\geq
-\frac{1}{4{\MATHSF{N}}^2}$.
+X_1=\ov{h}^{-1}(X))=\frac{1}{2{\mathsf{N}}}$, proving that $\P(S_{x,\ell}\leq 2)\geq
+\frac{1}{4{\mathsf{N}}^2}$.

 \end{itemize}
 
 
 \end{itemize}