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:E1,+ tel que PwCw, avec C0, et soit w l’espace des fonctions f mesurables de E dans telles que f w =supw(x) -1 f(x),xE<+. Nous démontrons que P est quasi-compact sur ( w ,· w ) si et seulement si, pour tout f w , (1 n k=1 n P k f) n contient une sous-suite convergeant dans w vers Πf= 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 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:E1,+ such that PwCw with C0, and let w be the space of measurable functions on E satisfying f w =supw(x) -1 f(x),xE<+. We prove that P is quasi-compact on ( w ,· w ) if and only if, for all f w , (1 n k=1 n P k f) n contains a subsequence converging in w to Πf= 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 w is finite-dimensional, uniform ergodicity is equivalent to mean 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},
     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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