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Generalized eigenfunctions of Markov kernels and application to the convergence rate of discrete random walks

Denis Guibourg 1, Loïc Hervé 1, James Ledoux 1

(2012-06-07)

Let $(X_n)_{n\in\N}$ be a Markov chain on a measurable space $\X$ with transition kernel $P$ and let $V:\X\r[1,+\infty)$. Under a weak drift condition, the size of generalized eigenfunctions of $P$ is estimated, where $P$ is here considered as a linear bounded operator on the weighted-supremum space $\cB_V$ associated with $V$. Then combining this result and quasi-compactness arguments enables us to derive upper bounds for the geometric rate of convergence of $(X_n)_{n\in\N}$ to its invariant probability measure in operator norm on $\cB_V$. Applications to discrete Markov random walks are presented.

1:	Institut de Recherche Mathématique de Rennes (IRMAR)
	CNRS : UMR6625 – Université de Rennes 1 – École normale supérieure de Cachan - ENS Cachan – Institut National des Sciences Appliquées (INSA) : - RENNES – Université de Rennes II - Haute Bretagne

Research team: Théorie ergodique Statistique

Subject

:

Mathematics/Probability

Keyword(s): Geometric ergodicity – quasi-compactness – Drift condition – Birth-and -Death Markov chains.

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From: Loïc Hervé <>

Submitted on: Thursday, 7 June 2012 18:03:54

Updated on: Friday, 8 June 2012 08:59:39