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\newcommand {\tv}[1] {\lVert #1 \rVert_{\rm TV}}
\def \top {1.8}
\def \topt {2.3}
\def \P {\mathbb{P}}
\def \ov {\overline}
\def \ts {\tau_{\rm stop}}


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\title{XXX}

\begin{document}

\author{Jean-François Couchot, Christophe Guyeux, Pierre-Cyrile Heam}
\address{Institut FEMTO-ST, Université de Franche-Comté, Belfort, France}

\author{Sylvain Contassot-Vivier}
\address{Loria - UMR 7503, Université de Lorraine, Nancy, France}

\date{...}

\begin{abstract}
This   paper  extends   the  results   presented  in~\cite{bcgr11ip}
and~\cite{chgw14oip} by using the \emph{chaotic} updating mode, in the sense
of F.  Robert~\cite{Robert}.  In this mode, several components of the system
may be  updated at each iteration.   At the theoretical level,  we show that
    the properties of  chaos and uniformity of the  obtained PRNG are preserved.
    At  the practical level,  we show  that the  algorithm that  builds strongly
    connected  iteration graphs,  with doubly  stochastic Markov  matrix,  has a
    reduced mixing time.
\end{abstract}

\section{Introduction}
%\input{intro}

 \section{\uppercase{Preliminaries}}\label{sec:preliminaries}
\input{preliminaries}


\section{Proof Of Chaos}
\input{chaos}


\section{Random walk on the modified Hypercube}
\input{stopping}

% Donner la borne du stopping time quand on marche dedans (nouveau). 
% Énoncer le problème de la taille de cette borne (elle est certes finie, mais grande). 




\section{Quality study of the strategy}
%6) Se pose alors la question de comment générer une stratégie de "bonne qualité". Par exemple, combien de générateurs aléatoires embarquer ? (nouveau)


\section{Application to Pseudorandom Number Generation}
%\input{prng}

\section{Conclusion}
%\input{conclusion}

%\acknowledgements{...}

\bibliographystyle{alpha}
\bibliography{biblio}

\end{document}
