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

Barron, Andrew R.

Ann. Probab., Tome 13 (1985) no. 4, p. 1292-1303 / Harvested from Project Euclid

Résumé

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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Publié le : 1985-11-14
Classification:  Shannon-McMillan-Breiman theorem,  Moy-Perez theorem,  asymptotic equipartition property,  martingale inequalities,  entropy,  information,  ergodic theorems,  asymptotically mean stationary,  28D05,  94A17,  62B10,  60F15,  60G10,  60G42,  28D20