We study the class $C_\pi$ of probability measures invariant with respect to the shift transformation on $K^\mathbb{Z}$ (where $K$ is a finite set of integers) which satisfies the Chapman-Kolmogorov equation for a given stochastic matrix $\Pi$. We construct a dense subset of measures in $C_\pi$ distinct from the Markov measure. When $\Pi$ is irreducible and aperiodic, these measures are ergodic but not weakly mixing. We show that the set of measures with infinite memory is $G_\delta$ dense in $C_\pi$ and that the Markov measure is the unique measure which maximizes the Kolmogorov-Sinai (K-S) entropy in $C_\pi$. We give examples of ergodic measures in $C_\pi$ with zero entropy.
@article{1176988618,
author = {Courbage, M. and Hamdan, D.},
title = {Chapman-Kolmogorov Equation for Non-Markovian Shift-Invariant Measures},
journal = {Ann. Probab.},
volume = {22},
number = {4},
year = {1994},
pages = { 1662-1677},
language = {en},
url = {http://dml.mathdoc.fr/item/1176988618}
}
Courbage, M.; Hamdan, D. Chapman-Kolmogorov Equation for Non-Markovian Shift-Invariant Measures. Ann. Probab., Tome 22 (1994) no. 4, pp. 1662-1677. http://gdmltest.u-ga.fr/item/1176988618/