A simple proof of the Moy-Perez generalization of the Shannon-McMillan theorem

The Shannon-McMillan Theorem of Information Theory has been generalized by Moy and Perez.The purpose of this paper is to give a simple proof of this generalization.1* Introduction* Let T be either the semigroup of nonnegative integers N or nonnegative real numbers R + .Let ^ = {17*: te T} be a semigroup of measurable mappings from a given measurable space (Ω, ^~) to itself.(We suppose U° is the identity map.)If X o is a measurable mapping from the space (Ω, JP") to another space, let (X t :te T) be the process generated by <%r; that is, X t = X Q -U\ t e T. If α, b 6 Γ, a ^ b, let J? ah denote the sub-sigmafield of ^ generated by the mappings {X t : te T, a ^t ^ &}.Let P, Q be probability measures on ^\ let P a b(Qab) be the restriction of P(Q) to _^δ.We suppose that P ot is absolutely continuous with respect to Q Oί , te T. Then the Radon-Nikodym derivatives f st = st exist, s^t.We assume that the entropies H st = \ st dP, s <^t, are all finite.(We use natural logarithms.)It is known that (1) H ot is a nonnegative, nondecreasing function of t ([6], p. 54); and(2) If | i | l denotes the L) norm, then ίi log (frjfst) II ^ H ru -H st + 1, r ^ s ^ t ^ u , ([6], inequality (2.4.10), and p. 54). The Moy-Perez result.The following generalization of the Shannon-McMillan Theorem was proved by Moy [4] for the case T = N, and by Perez [5], for the case T = R + .THEOREM.Let (X t :te T) be a stationary process with respect to P and a Markov process with stationary transition probabilities with respect to Q.If the sequence {n^Honi n = 1, 2, •} is bounded above, then the functions {t~~ι ot '. t > 0, te T} converge as t~+ °o in L) to a function h which is invariant with respect to ^ that is, h W = h, te T.