Soit un noyau markovien sur un espace mesurable muni d’une tribu à base dénombrable, soit tel que , avec , et soit l’espace des fonctions mesurables de dans telles que . Nous démontrons que est quasi-compact sur si et seulement si, pour tout , contient une sous-suite convergeant dans vers , où est une fonction mesurable positive bornée sur et une probabilité sur . En particulier, quand le sous-espace de constitué des fonctions -invariantes est de dimension finie, la convergence uniforme des moyennes est équivalente à la convergence ponctuelle.
Let be a Markov kernel on a measurable space with countably generated -algebra, let such that with , and let be the space of measurable functions on satisfying . We prove that is quasi-compact on if and only if, for all , contains a subsequence converging in to , where the ’s are non-negative bounded measurable functions on and the ’s are probability distributions on . In particular, when the space of -invariant functions in is finite-dimensional, uniform ergodicity is equivalent to mean ergodicity.
Mots clés : Markov kernel, quasi-compactness, mean ergodicity, geometrical ergodicity
@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}, pages = {1090--1095}, publisher = {Gauthier-Villars}, volume = {44}, number = {6}, year = {2008}, doi = {10.1214/07-AIHP145}, mrnumber = {2469336}, zbl = {1186.37014}, language = {en}, url = {http://www.numdam.org/articles/10.1214/07-AIHP145/} }
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 SP - 1090 EP - 1095 VL - 44 IS - 6 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/07-AIHP145/ DO - 10.1214/07-AIHP145 LA - en ID - AIHPB_2008__44_6_1090_0 ER -
%0 Journal Article %A Hervé, Loïc %T Quasi-compactness and mean ergodicity for Markov kernels acting on weighted supremum normed spaces %J Annales de l'I.H.P. Probabilités et statistiques %D 2008 %P 1090-1095 %V 44 %N 6 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/07-AIHP145/ %R 10.1214/07-AIHP145 %G en %F 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/articles/10.1214/07-AIHP145/
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