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A sandwich proof of the Shannon-McMillan-Breiman theorem. (English) Zbl 0653.28013
Let $$X_ 0,X_ 1,..$$. be a stationary ergodic process with Polish state space S. Let M be a reference measure on the space of all one-sided infinite sequences from S. Suppose that M is finite order Markov with stationary transition kernel. Further, suppose that for each positive integer n, the distribution $$P_ n$$ of $$(X_ 0,X_ 1,...,X_{n-1})$$ is absolutely continuous with respect to the marginal distribution $$M_ n$$ that one obtains from M by projecting on the first n coordinates. Let $$\{f_ n\}$$ be the sequence of random variables $$f_ n=(dP_ n/dM_ n)(X_ 0,x_ 1,...,X_{n-1}),\quad n=1,2,...\quad.$$ It is known that the sequence $$\{n^{-1} \log f_ n\}$$ converges almost surely. (This result is a generalization of the Shannon-McMillan-Breiman theorem.) The authors provide a new proof of this result which involves an ingenious “sandwiching” of the limit superior and limit inferior of $$\{n^{-1} \log f_ n\}$$ between two random variables that have the same expected value.
Reviewer: J.C.Kieffer

##### MSC:
 28D05 Measure-preserving transformations 94A17 Measures of information, entropy 60F15 Strong limit theorems
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