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

Christophe Cuny; Michael Lin

Annales de l'I.H.P. Probabilités et statistiques (2009)

  • Volume: 45, Issue: 3, page 710-733
  • ISSN: 0246-0203

Abstract

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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.

How to cite

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Cuny, Christophe, and Lin, Michael. "Pointwise ergodic theorems with rate and application to the CLT for Markov chains." Annales de l'I.H.P. Probabilités et statistiques 45.3 (2009): 710-733. <http://eudml.org/doc/78040>.

@article{Cuny2009,
abstract = {Let T be Dunford–Schwartz operator on a probability space (Ω, μ). For f∈Lp(μ), p&gt;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.},
author = {Cuny, Christophe, Lin, Michael},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {ergodic theorems with rates; central limit theorem for Markov chains; Dunford–Schwartz operators; probability preserving transformations; Dunford-Schwartz operators},
language = {eng},
number = {3},
pages = {710-733},
publisher = {Gauthier-Villars},
title = {Pointwise ergodic theorems with rate and application to the CLT for Markov chains},
url = {http://eudml.org/doc/78040},
volume = {45},
year = {2009},
}

TY - JOUR
AU - Cuny, Christophe
AU - Lin, Michael
TI - Pointwise ergodic theorems with rate and application to the CLT for Markov chains
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2009
PB - Gauthier-Villars
VL - 45
IS - 3
SP - 710
EP - 733
AB - Let T be Dunford–Schwartz operator on a probability space (Ω, μ). For f∈Lp(μ), p&gt;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.
LA - eng
KW - ergodic theorems with rates; central limit theorem for Markov chains; Dunford–Schwartz operators; probability preserving transformations; Dunford-Schwartz operators
UR - http://eudml.org/doc/78040
ER -

References

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