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

Hervé, Loïc

Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 6, pp. 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 $({ℬ}_w,\parallel ·{\parallel }_w)$ 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 $({ℬ}_w,\parallel ·{\parallel }_w)$ 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.

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