Pointwise ergodic theorems with rate and application to the CLT for Markov chains

Cuny, Christophe; Lin, Michael

Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 3, pp. 710-733.

Soit T un opérateur de Dunford-Schwartz sur un espace de probabilité (Ω, μ). Pour f∈L^p(μ), p>1, nous obtenons des théorèmes ergodiques du type (1/n^1/p)∑_k=1ⁿT^kf→0 μ-p.s. sous des conditions portant sur la croissance de ‖∑_k=1ⁿT^kf‖_p. Lorsque T est induit par une transformation préservant la mesure et que p=2, nous obtenons de meilleurs résultats. Ces derniers sont alors utilisés pour obtenir le théorème central limite «quenched» pour les sommes partielles associées aux fonctionnelles de chaînes de Markov stationnaires et ergodiques. Nous améliorons ainsi des résultats antérieurs de Derriennic-Lin et Wu-Woodroofe.

Let T be Dunford-Schwartz operator on a probability space (Ω, μ). For f∈L^p(μ), p>1, we obtain growth conditions on ‖∑_k=1ⁿT^kf‖_p which imply that (1/n^1/p)∑_k=1ⁿT^kf→0 μ-a.e. In the particular case that p=2 and T is the isometry induced by a probability preserving transformation we get better results than in the general case; these are used to obtain a quenched central limit theorem for additive functionals of stationary ergodic Markov chains, which improves those of Derriennic-Lin and Wu-Woodroofe.

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