A Sandwich Proof of the Shannon-McMillan-Breiman Theorem

Let \Xₜ\ be a stationary ergodic process with distribution P admitting densities p (x₀, , x₍-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^-1 p (X₀, , X₍-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.