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) no. 6, p. 1090-1095
Soit $P$ un noyau markovien sur un espace mesurable $E$ muni d’une tribu à base dénombrable, soit $w:E\to \left[1,+\infty \right[$ tel que $Pw\le Cw$, avec $C\ge 0$, et soit ${ℬ}_{w}$ l’espace des fonctions $f$ mesurables de $E$ dans $ℂ$ telles que ${∥f∥}_{w}=sup\left\{w{\left(x\right)}^{-1}\left|f\left(x\right)\right|,x\in E\right\}<+\infty$. Nous démontrons que $P$ est quasi-compact sur $\left({ℬ}_{w},\parallel ·{\parallel }_{w}\right)$ si et seulement si, pour tout $f\in {ℬ}_{w}$, ${\left(\frac{1}{n}{\sum }_{k=1}^{n}{P}^{k}f\right)}_{n}$ contient une sous-suite convergeant dans ${ℬ}_{w}$ vers $\Pi f={\sum }_{i=1}^{d}{\mu }_{i}\left(f\right){v}_{i}$, où ${v}_{i}$ est une fonction mesurable positive bornée sur $E$ et ${\mu }_{i}$ 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 $\sigma$-algebra, let $w:E\to \left[1,+\infty \right[$ such that $Pw\le Cw$ with $C\ge 0$, and let ${ℬ}_{w}$ be the space of measurable functions on $E$ satisfying ${∥f∥}_{w}=sup\left\{w{\left(x\right)}^{-1}\left|f\left(x\right)\right|,x\in E\right\}<+\infty$. We prove that $P$ is quasi-compact on $\left({ℬ}_{w},\parallel ·{\parallel }_{w}\right)$ if and only if, for all $f\in {ℬ}_{w}$, ${\left(\frac{1}{n}{\sum }_{k=1}^{n}{P}^{k}f\right)}_{n}$ contains a subsequence converging in ${ℬ}_{w}$ to $\Pi f={\sum }_{i=1}^{d}{\mu }_{i}\left(f\right){v}_{i}$, where the ${v}_{i}$’s are non-negative bounded measurable functions on $E$ and the ${\mu }_{i}$’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.
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},
publisher = {Gauthier-Villars},
volume = {44},
number = {6},
year = {2008},
pages = {1090-1095},
doi = {10.1214/07-AIHP145},
zbl = {1186.37014},
mrnumber = {2469336},
language = {en},
url = {http://www.numdam.org/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) no. 6, pp. 1090-1095. doi : 10.1214/07-AIHP145. http://www.numdam.org/item/AIHPB_2008__44_6_1090_0/

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