Évènement

The jury consists of

• Perla Sousi — Professor, University of Cambridge — Reviewer
• Laurent Saloff-Coste — Professor, Cornell University — Reviewer
• Nina Gantert — Professor, Technical University of Munich — Chair of the Jury
• Laurent Miclo — Senior Researcher, Toulouse School of Economics and Institute of Mathematics — Examiner
• Fabienne Castell — Professor, Aix-Marseille University — Examiner
• Bruno Schapira — Professor, University of Lyon 1 — Examiner
• Ruojun Huang — Postdoctoral Researcher, Scuola Normale Superiore di Pisa — Examiner
• Pierre Mathieu — Professor, Aix-Marseille University — Thesis Supervisor

In this thesis, we study the merging property of a discrete-time Markov chain on a countable space. Merging refers to the chain’s ability to forget its initial distribution. After an introductory chapter where we present the basic concepts and necessary vocabulary to understand our results on merging, we dedicate the second chapter to a sufficient condition for establishing this merging. We assume conditions ensuring irreducibly, aperiodicity, and the tightness of a sequence of laws with a proper initial probability measure. We then prove that the chain merges. Let ((K_t)_{t geq 1}) be a sequence of Markov transition operators on the same discrete set (V). Let (mu_t^x) denote the law at time (t) starting from (x), driven by the sequence ((K_t)_{t geq 1}), and let (d_{TV}) be the classical total variation distance. Let (rho) be a proper probability measure on (V). Denote by ((rho_t)_{t geq 0}) the sequence of laws driven by the sequence ((K_t)_{t geq 1}) with the initial law (rho). If the sequence ((rho_t)_{t geq 0}) is tight, then:
[
forall (x,y) in V, quad lim_{t to +infty} d_{TV}(mu_t^x, mu_t^y) = 0,.
]
This result is proved using an original tool: the accompanying sets process. This random object is derived from the evolving sets introduced by B. Morris and Y. Peres. It also allows us to provide a result within the framework of ratio theorems, but under the hypothesis that there exists a finite subset (A) of (V) such that the limit inferior of the sequence ((rho_t(A))_{t geq 0}) is strictly positive, instead of the usual tightness assumption. Additionally, in this same chapter, we study the convergence of the laws when the operators or invariant measures converge pointwise.