Chapman-Kolmogorov Equation for Non-Markovian Shift-Invariant Measures

Abstract

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.