Upcrossing inequalities for stationary sequences and applications

Michael Hochman

doi:10.1214/09-AOP460

Upcrossing inequalities for stationary sequences and applications

Michael Hochman^*

^*Corresponding author for this work

Einstein Institute of Mathematics

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

19 Scopus citations

Abstract

For arrays (S_i,j)_1≤i≤j of random variables that are stationary in an appropriate sense, we show that the fluctuations of the process (S_1,n)^∞_n=1 can be bounded in terms of a measure of the "mean subadditivity" of the process (S_i,j)_1≤i≤j. We derive universal upcrossing inequalities with exponential decay for Kingman's subadditive ergodic theorem, the Shannon-MacMillan-Breiman theorem and for the convergence of the Kolmogorov complexity of a stationary sample.

Original language	English
Pages (from-to)	2135-2149
Number of pages	15
Journal	Annals of Probability
Volume	37
Issue number	6
DOIs	https://doi.org/10.1214/09-AOP460
State	Published - Nov 2009

Keywords

Almost everywhere convergence
Ergodic theorem
Kolmogorov complexity
Shannon-McMillan-Breiman theorem
Upcrossing inequalities entropy

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10.1214/09-AOP460

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AB - For arrays (Si,j)1≤i≤j of random variables that are stationary in an appropriate sense, we show that the fluctuations of the process (S1,n)∞n=1 can be bounded in terms of a measure of the "mean subadditivity" of the process (Si,j)1≤i≤j. We derive universal upcrossing inequalities with exponential decay for Kingman's subadditive ergodic theorem, the Shannon-MacMillan-Breiman theorem and for the convergence of the Kolmogorov complexity of a stationary sample.

KW - Almost everywhere convergence

KW - Ergodic theorem

KW - Kolmogorov complexity

KW - Shannon-McMillan-Breiman theorem

KW - Upcrossing inequalities entropy

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JO - Annals of Probability

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Upcrossing inequalities for stationary sequences and applications

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Keywords

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