Open Access
October, 1980 Asymptotically Mean Stationary Measures
Robert M. Gray, J. C. Kieffer
Ann. Probab. 8(5): 962-973 (October, 1980). DOI: 10.1214/aop/1176994624


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.


Download Citation

Robert M. Gray. J. C. Kieffer. "Asymptotically Mean Stationary Measures." Ann. Probab. 8 (5) 962 - 973, October, 1980.


Published: October, 1980
First available in Project Euclid: 19 April 2007

zbMATH: 0447.28014
MathSciNet: MR586779
Digital Object Identifier: 10.1214/aop/1176994624

Primary: 28A65
Secondary: 60G10 , 94A15

Keywords: asymptotically mean stationary , ergodic theorems , Shannon-McMillan-Breiman theorem

Rights: Copyright © 1980 Institute of Mathematical Statistics

Vol.8 • No. 5 • October, 1980
Back to Top