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hal-00705523, version 1

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

Denis Guibourg () 1, Loïc Hervé () 1, James Ledoux () 1

• 1:  Institut de Recherche Mathématique de Rennes (IRMAR)
• http://irmar.univ-rennes1.fr/
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 France
• Available versions :  v1 (2012-06-07) v2 (2013-04-02)
• ### Bibliographic reference

• Type of document: Documents without publication reference (Preprint)
• Subject: Mathematics/Probability
• Title: Generalized eigenfunctions of Markov kernels and application to the convergence rate of discrete random walks
• Abstract: 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.
• Fulltext language: English
• Keyword(s): Geometric ergodicity – quasi-compactness – Drift condition – Birth-and -Death Markov chains.
• Classification: 60J10; 47B07
• Internal note: 2012-50

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• hal-00705523, version 1
• oai:hal.archives-ouvertes.fr:hal-00705523
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• Submitted on: Thursday, 7 June 2012 18:03:54
• Updated on: Friday, 8 June 2012 08:59:39