## The strong ergodic theorem for densities: Generalized Shannon-McMillan- Breiman theorem.(English)Zbl 0608.94001

Let $$(X_ 1,X_ 2,...)$$ be a stationary process with probability densities $$f(X_ 1,X_ 2,...,X_ n)$$ 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$${}_ 1,X_ 2,...,X_ n)$$ 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.

### MSC:

 94A17 Measures of information, entropy 28D05 Measure-preserving transformations 60G42 Martingales with discrete parameter 62B10 Statistical aspects of information-theoretic topics 60F15 Strong limit theorems 60G10 Stationary stochastic processes
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