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| HAL : hal-00632580, version 5 |
| arXiv : 1110.3240 |
| Fiche détaillée |  Récupérer au format |
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| Versions disponibles : | v1 (14-10-2011) | v2 (17-10-2011) | v3 (02-03-2012) | v4 (25-05-2012) | v5 (12-06-2012) |
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| Quasi-compactness of Markov kernels on weighted-supremum spaces and geometrical ergodicity |
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| Denis Guibourg 1Loïc Hervé 1 |
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| (29/02/2012) |
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| 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. |
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| 1 : | Institut de Recherche Mathématique de Rennes (IRMAR) |
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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 |
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| Théorie ergodique Statistique |
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| Domaine | : | Mathématiques/Probabilités |
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| 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 | |
| Contributeur : James Ledoux | |
| Soumis le : Mardi 12 Juin 2012, 18:25:16 | |
| Dernière modification le : Mardi 12 Juin 2012, 21:05:08 | |
