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

Loïc Hervé

Annales de l'I.H.P. Probabilités et statistiques (2008)

  • Volume: 44, Issue: 6, page 1090-1095
  • ISSN: 0246-0203

Abstract

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Let P be a Markov kernel on a measurable space E with countably generated σ-algebra, let w:E→[1, +∞[ such that Pw≤Cw with C≥0, and let ℬ w be the space of measurable functions onE satisfying ‖f‖w=sup{w(x)−1|f(x)|, x∈E}<+∞. We prove that Pis quasi-compact on ( ℬ w , ∥ · ∥ w ) if and only if, for all f ∈ ℬ w , ( 1 n ∑ k = 1 n P k f ) n contains a subsequence converging in ℬ w toΠf=∑di=1μi(f)vi, where the vi’s are non-negative bounded measurable functions on E and the μi’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.

How to cite

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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 44.6 (2008): 1090-1095. <http://eudml.org/doc/78004>.

@article{Hervé2008,
abstract = {Let P be a Markov kernel on a measurable space E with countably generated σ-algebra, let w:E→[1, +∞[ such that Pw≤Cw with C≥0, and let $\mathcal \{B\}_\{w\}$ be the space of measurable functions onE satisfying ‖f‖w=sup\{w(x)−1|f(x)|, x∈E\}&lt;+∞. We prove that Pis quasi-compact on $(\mathcal \{B\}_\{w\},\Vert \cdot \Vert _\{w\})$ if and only if, for all $f\in \mathcal \{B\}_\{w\}$, $(\frac\{1\}\{n\}\sum _\{k=1\}^\{n\}P^\{k\}f)_\{n\}$ contains a subsequence converging in $\mathcal \{B\}_\{w\}$ toΠf=∑di=1μi(f)vi, where the vi’s are non-negative bounded measurable functions on E and the μi’s are probability distributions on E. In particular, when the space of P-invariant functions in $\mathcal \{B\}_\{w\}$ is finite-dimensional, uniform ergodicity is equivalent to mean ergodicity.},
author = {Hervé, Loïc},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {Markov kernel; quasi-compactness; mean ergodicity; geometrical ergodicity; supremum normed spaces},
language = {eng},
number = {6},
pages = {1090-1095},
publisher = {Gauthier-Villars},
title = {Quasi-compactness and mean ergodicity for Markov kernels acting on weighted supremum normed spaces},
url = {http://eudml.org/doc/78004},
volume = {44},
year = {2008},
}

TY - JOUR
AU - Hervé, Loïc
TI - Quasi-compactness and mean ergodicity for Markov kernels acting on weighted supremum normed spaces
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2008
PB - Gauthier-Villars
VL - 44
IS - 6
SP - 1090
EP - 1095
AB - Let P be a Markov kernel on a measurable space E with countably generated σ-algebra, let w:E→[1, +∞[ such that Pw≤Cw with C≥0, and let $\mathcal {B}_{w}$ be the space of measurable functions onE satisfying ‖f‖w=sup{w(x)−1|f(x)|, x∈E}&lt;+∞. We prove that Pis quasi-compact on $(\mathcal {B}_{w},\Vert \cdot \Vert _{w})$ if and only if, for all $f\in \mathcal {B}_{w}$, $(\frac{1}{n}\sum _{k=1}^{n}P^{k}f)_{n}$ contains a subsequence converging in $\mathcal {B}_{w}$ toΠf=∑di=1μi(f)vi, where the vi’s are non-negative bounded measurable functions on E and the μi’s are probability distributions on E. In particular, when the space of P-invariant functions in $\mathcal {B}_{w}$ is finite-dimensional, uniform ergodicity is equivalent to mean ergodicity.
LA - eng
KW - Markov kernel; quasi-compactness; mean ergodicity; geometrical ergodicity; supremum normed spaces
UR - http://eudml.org/doc/78004
ER -

References

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