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.
Publié le : 1980-10-14
Classification:
Asymptotically mean stationary,
ergodic theorems,
Shannon-McMillan-Breiman theorem,
28A65,
94A15,
60G10
@article{1176994624,
author = {Gray, Robert M. and Kieffer, J. C.},
title = {Asymptotically Mean Stationary Measures},
journal = {Ann. Probab.},
volume = {8},
number = {6},
year = {1980},
pages = { 962-973},
language = {en},
url = {http://dml.mathdoc.fr/item/1176994624}
}
Gray, Robert M.; Kieffer, J. C. Asymptotically Mean Stationary Measures. Ann. Probab., Tome 8 (1980) no. 6, pp. 962-973. http://gdmltest.u-ga.fr/item/1176994624/