A simple proof of the Moy-Perez generalization of the Shannon-McMillan theorem
John Kieffer · Pacific Journal of Mathematics · 1974
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 0, te T} converge as t~+ °o in L\P) to a function h which is invariant with respect to ^ that is, h W = h, te T.To prove this theorem, Moy and Perez embedded the process (X t ) in a bilateral process (X t : -oo < t < °°), stationary with respect to P and Markov with respect to Q; Doob's Martingale Convergence 203