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November, 1985 The Strong Ergodic Theorem for Densities: Generalized Shannon-McMillan-Breiman Theorem
Andrew R. Barron
Ann. Probab. 13(4): 1292-1303 (November, 1985). DOI: 10.1214/aop/1176992813


Let $\{X_1, X_2,\cdots\}$ be a stationary process with probability densities $f(X_1, X_2,\cdots, X_n)$ with respect to Lebesgue measure or with respect to a Markov measure with a stationary transition measure. It is shown that the sequence of relative entropy densities $(1/n)\log f(X_1, X_2,\cdots, X_n)$ converges almost surely. This long-conjectured result extends the $L^1$ convergence obtained by Moy, Perez, and Kieffer and generalizes the Shannon-McMillan-Breiman theorem to nondiscrete processes. The heart of the proof is a new martingale inequality which shows that logarithms of densities are $L^1$ dominated.


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Andrew R. Barron. "The Strong Ergodic Theorem for Densities: Generalized Shannon-McMillan-Breiman Theorem." Ann. Probab. 13 (4) 1292 - 1303, November, 1985.


Published: November, 1985
First available in Project Euclid: 19 April 2007

zbMATH: 0608.94001
MathSciNet: MR806226
Digital Object Identifier: 10.1214/aop/1176992813

Primary: 28D05
Secondary: 28D20 , 60F15 , 60G10 , 60G42 , 62B10 , 94A17

Keywords: asymptotic equipartition property , asymptotically mean stationary , Entropy , ergodic theorems , Information , martingale inequalities , Moy-Perez theorem , Shannon-McMillan-Breiman theorem

Rights: Copyright © 1985 Institute of Mathematical Statistics

Vol.13 • No. 4 • November, 1985
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