Asymptotically Mean Stationary Measures

Abstract

Numerous properties are developed of measures that are asymptotically mean stationary with respect to a possibly nonsingular and noninvertible measurable transformation on a probability space. In particular, several necessary and sufficient conditions for the measure and transformation to satisfy the ergodic theorem are given, an asymptotic form of the Radon-Nikodym theorem for asymptotically dominated measures is developed, and the asymptotic behavior of the resulting Radon-Nikodym derivatives is described. As an application we prove a Shannon-McMillan-Breiman theorem for the case considered. Several examples are given to illustrate the results.