## The Annals of Probability

### 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.

#### Article information

Source
Ann. Probab., Volume 22, Number 3 (1994), 1662-1677.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176988618

Digital Object Identifier
doi:10.1214/aop/1176988618

Mathematical Reviews number (MathSciNet)
MR1303660

Zentralblatt MATH identifier
0824.60033

JSTOR
links.jstor.org

Subjects
Primary: 60G10: Stationary processes
Secondary: 28D05: Measure-preserving transformations

#### Citation

Courbage, M.; Hamdan, D. Chapman-Kolmogorov Equation for Non-Markovian Shift-Invariant Measures. Ann. Probab. 22 (1994), no. 3, 1662--1677. doi:10.1214/aop/1176988618. https://projecteuclid.org/euclid.aop/1176988618