941 articles – 1212 references  [version française]
 HAL: hal-00632580, version 5
 arXiv: 1110.3240
 Available versions: v1 (2011-10-14) v2 (2011-10-17) v3 (2012-03-02) v4 (2012-05-25) v5 (2012-06-12)
 Quasi-compactness of Markov kernels on weighted-supremum spaces and geometrical ergodicity
 Denis Guibourg 1, Loïc Hervé 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 ergodiqueStatistique
 Subject : Mathematics/Probability
 Keyword(s): Markov chain – drift condition – essential spectral radius – convergence rate
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 hal-00632580, version 5 http://hal.archives-ouvertes.fr/hal-00632580 oai:hal.archives-ouvertes.fr:hal-00632580 From: James Ledoux <> Submitted on: Tuesday, 12 June 2012 18:25:16 Updated on: Tuesday, 12 June 2012 21:05:08