Stochastic processes with variable length memory: Monge-Kantorovich problem, bootstrap and particle systems

Abstract

This project contains two principal parts which can be seen as a continuation and generalization of the PhD Thesis of the candidate. First, we will use the regenerative structure of the stochastic processes with unbounded variable length memory which have been introduced in the PhD Thesis \citep{gallo/2009} to solve two questions of actual probability theory. The first one is the Monge-Kantorovich problem for stationary laws of infinite memory chains. The second is the use of the regeneration scheme as a base to perform bootstrap for sotchastic chains of infinite memory. In a second part, we would like to find new criteria ensuring the existence and uniqueness of the stationary law for markovian particle system with infinite range, and in particular, an alternative to the traditional continuity assumption \citep*{galves/garcia/eva/2009}. Such an alternative can also be found in the notion of processes with variable range. Hence, a natural way would be to extend the results of the PhD Thesis of the candidate on perfect simulation for stochastic chains with variable length memory to the case of particle systems with variable range. (AU)