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

Christophe Cuny; Michael Lin

doi:10.1214/08-AIHP180

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

Christophe Cuny, Michael Lin

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

15 Scopus citations

Abstract

Let T be Dunford-Schwartz operator on a probability space (ω, μ). For f ε L^p(μ),p >1, we obtain growth conditions on || ∑ _k ⁿ =1 T ^kf ||p which imply that 1/n ¹/p ∑_kⁿ=1 T ^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.

Original language	English
Pages (from-to)	710-733
Number of pages	24
Journal	Annales de l'institut Henri Poincare (B) Probability and Statistics
Volume	45
Issue number	3
DOIs	https://doi.org/10.1214/08-AIHP180
State	Published - 1 Aug 2009

Keywords

Central limit theorem for Markov chains
Dunford-Schwartz operators
Ergodic theorems with rates
Probability preserving Transformations

ASJC Scopus subject areas

Statistics and Probability
Statistics, Probability and Uncertainty

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10.1214/08-AIHP180

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AU - Lin, Michael

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N2 - Let T be Dunford-Schwartz operator on a probability space (ω, μ). For f ε Lp(μ),p >1, we obtain growth conditions on || ∑ k n =1 T kf ||p which imply that 1/n 1/p ∑kn=1 T 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.

AB - Let T be Dunford-Schwartz operator on a probability space (ω, μ). For f ε Lp(μ),p >1, we obtain growth conditions on || ∑ k n =1 T kf ||p which imply that 1/n 1/p ∑kn=1 T 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.

KW - Central limit theorem for Markov chains

KW - Dunford-Schwartz operators

KW - Ergodic theorems with rates

KW - Probability preserving Transformations

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

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Keywords

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