Quasi-compactness and mean ergodicity for Markov kernels acting on weighted supremum normed spaces

Hervé, Loïc

Hervé, Loïc

Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008), p. 1090-1095 / Harvested from Numdam

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Résumé
Abstract

Soit $P$ un noyau markovien sur un espace mesurable $E$ muni d’une tribu à base dénombrable, soit $w : E \to [1, + \infty[$ tel que $P w \leq C w$ , avec $C \geq 0$ , et soit $ℬ_{w}$ l’espace des fonctions $f$ mesurables de $E$ dans $ℂ$ telles que ${∥f∥}_{w} = sup \{w {(x)}^{- 1} |f (x)|, x \in E\} < + \infty$ . Nous démontrons que $P$ est quasi-compact sur $(ℬ_{w}, ∥ \cdot ∥_{w})$ si et seulement si, pour tout $f \in ℬ_{w}$ , ${(\frac{1}{n} \sum_{k = 1}^{n} P^{k} f)}_{n}$ contient une sous-suite convergeant dans $ℬ_{w}$ vers $Π f = \sum_{i = 1}^{d} μ_{i} (f) v_{i}$ , où $v_{i}$ est une fonction mesurable positive bornée sur $E$ et $μ_{i}$ une probabilité sur $E$ . En particulier, quand le sous-espace de $ℬ_{w}$ constitué des fonctions $P$ -invariantes est de dimension finie, la convergence uniforme des moyennes est équivalente à la convergence ponctuelle.

Let $P$ be a Markov kernel on a measurable space $E$ with countably generated $σ$ -algebra, let $w : E \to [1, + \infty[$ such that $P w \leq C w$ with $C \geq 0$ , and let $ℬ_{w}$ be the space of measurable functions on $E$ satisfying ${∥f∥}_{w} = sup \{w {(x)}^{- 1} |f (x)|, x \in E\} < + \infty$ . We prove that $P$ is quasi-compact on $(ℬ_{w}, ∥ \cdot ∥_{w})$ if and only if, for all $f \in ℬ_{w}$ , ${(\frac{1}{n} \sum_{k = 1}^{n} P^{k} f)}_{n}$ contains a subsequence converging in $ℬ_{w}$ to $Π f = \sum_{i = 1}^{d} μ_{i} (f) v_{i}$ , where the $v_{i}$ ’s are non-negative bounded measurable functions on $E$ and the $μ_{i}$ ’s are probability distributions on $E$ . In particular, when the space of $P$ -invariant functions in $ℬ_{w}$ is finite-dimensional, uniform ergodicity is equivalent to mean ergodicity.

Publié le : 2008-01-01

MR 2469336 | Zbl 1186.37014

DOI : https://doi.org/10.1214/07-AIHP145
Classification: 37A30, 60J10

@article{AIHPB_2008__44_6_1090_0,
     author = {Herv\'e, Lo\"\i c},
     title = {Quasi-compactness and mean ergodicity for Markov kernels acting on weighted supremum normed spaces},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {44},
     year = {2008},
     pages = {1090-1095},
     doi = {10.1214/07-AIHP145},
     mrnumber = {2469336},
     zbl = {1186.37014},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2008__44_6_1090_0}
}

Hervé, Loïc. Quasi-compactness and mean ergodicity for Markov kernels acting on weighted supremum normed spaces. Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) pp. 1090-1095. doi : 10.1214/07-AIHP145. http://gdmltest.u-ga.fr/item/AIHPB_2008__44_6_1090_0/

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