X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rairo15.git/blobdiff_plain/7dd748e9a068c99c61fefdcbde3c24e4c9b74bf9..4da7584dffe58a5bbdcd406436a5b139ee4c13fd:/stopping.tex?ds=sidebyside

diff --git a/stopping.tex b/stopping.tex
index d3b4f4d..9cc720c 100644
--- a/stopping.tex
+++ b/stopping.tex
@@ -139,10 +139,10 @@ such that  the distribution of $X_\tau$ is $\pi$:
 $$\P_X(X_\tau=Y)=\pi(Y).$$
 
 
-\begin{Theo}
+\begin{thrm}
 If $\tau$ is a strong stationary time, then $d(t)\leq \max_{X\in\Bool^{\mathsf{N}}}
 \P_X(\tau > t)$.
-\end{Theo}
+\end{thrm}
 
 
 %Let $\Bool^n$ be the set of words of length $n$. 
@@ -182,9 +182,9 @@ $(X,\ov{h}(X)) \in E$ and $X\oplus\ov{h}(X)=0^{{\mathsf{N}}-h(X)}10^{h(X)-1}$.
 The function $\ov{h}$ is said {\it square-free} if for every $X\in \Bool^{\mathsf{N}}$,
 $\ov{h}(\ov{h}(X))\neq X$. 
 
-\begin{Lemma}\label{lm:h}
+\begin{lmm}\label{lm:h}
 If $\ov{h}$ is bijective and square-free, then $h(\ov{h}^{-1}(X))\neq h(X)$.
-\end{Lemma}
+\end{lmm}
 
 \begin{proof}
 Let $\ov{h}$ be bijective.
@@ -233,9 +233,9 @@ are fair. The integer $\ts$ is a randomized stopping time for
 the Markov chain $(X_t)$.
 
 
-\begin{Lemma}
+\begin{lmm}
 The integer $\ts$ is a strong stationary time.
-\end{Lemma}
+\end{lmm}
 
 \begin{proof}
 Let $\tau_\ell$ be the first time that $\ell$ is fair. The random variable
@@ -252,10 +252,10 @@ the same probability. Therefore,  for $t\geq \tau_\ell$, the
 $\ell$-th bit of $X_t$ is $0$ or $1$ with the same probability, proving the
 lemma.\end{proof}
 
-\begin{Theo} \label{prop:stop}
+\begin{thrm} \label{prop:stop}
 If $\ov{h}$ is bijective and square-free, then
 $E[\ts]\leq 8{\mathsf{N}}^2+ {\mathsf{N}}\ln ({\mathsf{N}}+1)$. 
-\end{Theo}
+\end{thrm}
 
 For each $X\in \Bool^{\mathsf{N}}$ and $\ell\in\llbracket 1,{\mathsf{N}}\rrbracket$, 
 let $S_{X,\ell}$ be the
@@ -268,11 +268,11 @@ $$S_{X,\ell}=\min \{t \geq 1\mid h(X_{t-1})\neq \ell\text{ and }Z_t=(\ell,.)\tex
 $$\lambda_h=\max_{X,\ell} S_{X,\ell}.$$
 
 
-\begin{Lemma}\label{prop:lambda}
+\begin{lmm}\label{prop:lambda}
 If $\ov{h}$ is a square-free bijective function, then the inequality 
 $E[\lambda_h]\leq 8{\mathsf{N}}^2$ is established.
 
-\end{Lemma}
+\end{lmm}
 
 \begin{proof}
 For every $X$, every $\ell$, one has $\P(S_{X,\ell})\leq 2)\geq
@@ -318,9 +318,9 @@ which concludes the proof.
 Let $\ts^\prime$ be the first time that there are exactly ${\mathsf{N}}-1$ fair
 elements. 
 
-\begin{Lemma}\label{lm:stopprime}
+\begin{lmm}\label{lm:stopprime}
 One has $E[\ts^\prime]\leq {\mathsf{N}} \ln ({\mathsf{N}}+1).$
-\end{Lemma}
+\end{lmm}
 
 \begin{proof}
 This is a classical  Coupon Collector's like problem. Let $W_i$ be the