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The Strong Ergodic Theorem for Densities: Generalized Shannon-McMillan-Breiman Theorem

Andrew R. Barron
The Annals of Probability
Vol. 13, No. 4 (Nov., 1985), pp. 1292-1303
Stable URL: http://www.jstor.org/stable/2244180
Page Count: 12
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The Strong Ergodic Theorem for Densities: Generalized Shannon-McMillan-Breiman Theorem
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Abstract

Let {X1, X2,⋯} be a stationary process with probability densities f(X1, X2,⋯, Xn) 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(X1, X2,⋯, Xn) converges almost surely. This long-conjectured result extends the L1 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 L1 dominated.

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