The Annals of Probability

Upcrossing inequalities for stationary sequences and applications

Michael Hochman

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For arrays (Si,j)1≤ij 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≤ij. 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.

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Ann. Probab., Volume 37, Number 6 (2009), 2135-2149.

First available in Project Euclid: 16 November 2009

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Primary: 37A30: Ergodic theorems, spectral theory, Markov operators {For operator ergodic theory, see mainly 47A35} 37A35: Entropy and other invariants, isomorphism, classification 60G10: Stationary processes 60G17: Sample path properties 94A17: Measures of information, entropy 68Q30: Algorithmic information theory (Kolmogorov complexity, etc.) [See also 03D32]

Ergodic theorem Shannon–McMillan–Breiman theorem upcrossing inequalities entropy Kolmogorov complexity almost everywhere convergence


Hochman, Michael. Upcrossing inequalities for stationary sequences and applications. Ann. Probab. 37 (2009), no. 6, 2135--2149. doi:10.1214/09-AOP460.

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