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A Sandwich Proof of the Shannon-McMillan-Breiman Theorem

Paul H. Algoet and Thomas M. Cover
The Annals of Probability
Vol. 16, No. 2 (Apr., 1988), pp. 899-909
Stable URL: http://www.jstor.org/stable/2243846
Page Count: 11
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A Sandwich Proof of the Shannon-McMillan-Breiman Theorem
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Abstract

Let {Xt} be a stationary ergodic process with distribution P admitting densities p(x0,..., xn-1) relative to a reference measure M that is finite order Markov with stationary transition kernel. Let IM(P) denote the relative entropy rate. Then n-1log p(X0,..., Xn-1) → IM(P) a.s. (P). We present an elementary proof of the Shannon-McMillan-Breiman theorem and the preceding generalization, obviating the need to verify integrability conditions and also covering the case IM(P) = ∞. A sandwich argument reduces the proof to direct applications of the ergodic theorem.

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