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arXiv: 1110.3240

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Quasi-compactness of Markov kernels on weighted-supremum spaces and geometrical ergodicity

Denis Guibourg 1, Loïc Hervé 1, James Ledoux 1

(2012-02-29)

Let $P$ be a Markov kernel on a measurable space $\X$ and let $V:\X\r[1,+\infty)$. We provide various assumptions, based on drift conditions, under which $P$ is quasi-compact on the weighted-supremum Banach space $(\cB_V,\|\cdot\|_V)$ of all the measurable functions $f : \X\r\C$ such that $\|f\|_V := \sup_{x\in \X} |f(x)|/V(x) < \infty$. Furthermore we give bounds for the essential spectral radius of $P$. Under additional assumptions, these results allow us to derive the convergence rate of $P$ on $\cB_V$, that is the geometric rate of convergence of the iterates $P^n$ to the stationary distribution in operator norm. Applications to discrete Markov kernels and to iterated function systems 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): Markov chain – drift condition – essential spectral radius – convergence rate

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From: James Ledoux <>

Submitted on: Tuesday, 12 June 2012 18:25:16

Updated on: Tuesday, 12 June 2012 21:05:08