$$\P_X(X_\tau=Y)=\pi(Y).$$
-\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 $n$.
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.
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
$\ell$-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
$$\lambda_h=\max_{X,\ell} S_{X,\ell}.$$
-\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
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
