The Annals of Probability

Un Theoreme Ergodique Presque Sous-Additif

Yves Derriennic

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Abstract

The two following results are proved. Given $(\Omega, \mathscr{F}, \mu, T)$ where $T$ is a measurable transformation preserving the probability measure $\mu$, given a sequence $f_n$ of integrable functions such that $\int (f_{n+k} - f_n - T^nf_k)^+ d\mu \leq c_k \text{with} \lim_k \frac{1}{k} c_k = 0,$ then $(1/n) f_n$ is converging in $L^1$-norm. If, furthermore, $f_{n+k} - f_n - T^nf_k \leq T^nh_k$ with $h_k$ a sequence of positive functions whose integrals are bounded, then $(1/n) f_n$ is also converging a.e. From this extension of Kingman's subadditive ergodic theorem, the Shannon-McMillan-Breiman theorem follows at once.

Article information

Source
Ann. Probab., Volume 11, Number 3 (1983), 669-677.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176993511

Digital Object Identifier
doi:10.1214/aop/1176993511

Mathematical Reviews number (MathSciNet)
MR704553

Zentralblatt MATH identifier
0586.28014

JSTOR
links.jstor.org

Subjects
Primary: 28D05: Measure-preserving transformations
Secondary: 60G10: Stationary processes

Keywords
Ergodic theorem a.e. convergence $L^1$-convergence subadditive sequence almost subadditive sequence entropy

Citation

Derriennic, Yves. Un Theoreme Ergodique Presque Sous-Additif. Ann. Probab. 11 (1983), no. 3, 669--677. doi:10.1214/aop/1176993511. https://projecteuclid.org/euclid.aop/1176993511


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