Let T be Dunford–Schwartz operator on a probability space (Ω, μ). For f∈Lp(μ), p>1, we obtain growth conditions on ‖∑k=1nTkf‖p which imply that (1/n1/p)∑k=1nTkf→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.
Publié le : 2009-08-15
Classification:
Ergodic theorems with rates,
Central limit theorem for Markov chains,
Dunford–Schwartz operators,
Probability preserving transformations,
60F05,
60J05,
37A30,
37A05,
47A35,
37A50
@article{1249391381,
author = {Cuny, Christophe and Lin, Michael},
title = {Pointwise ergodic theorems with rate and application to the CLT for Markov chains},
journal = {Ann. Inst. H. Poincar\'e Probab. Statist.},
volume = {45},
number = {1},
year = {2009},
pages = { 710-733},
language = {en},
url = {http://dml.mathdoc.fr/item/1249391381}
}
Cuny, Christophe; Lin, Michael. Pointwise ergodic theorems with rate and application to the CLT for Markov chains. Ann. Inst. H. Poincaré Probab. Statist., Tome 45 (2009) no. 1, pp. 710-733. http://gdmltest.u-ga.fr/item/1249391381/