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

M. Courbage and D. Hamdan
The Annals of Probability
Vol. 22, No. 3 (Jul., 1994), pp. 1662-1677
Stable URL: http://www.jstor.org/stable/2245039
Page Count: 16
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Chapman-Kolmogorov Equation for Non-Markovian Shift-Invariant Measures
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Abstract

We study the class Cπ of probability measures invariant with respect to the shift transformation on KZ (where K is a finite set of integers) which satisfies the Chapman-Kolmogorov equation for a given stochastic matrix Π. We construct a dense subset of measures in Cπ distinct from the Markov measure. When Π is irreducible and aperiodic, these measures are ergodic but not weakly mixing. We show that the set of measures with infinite memory is Gδ dense in Cπ and that the Markov measure is the unique measure which maximizes the Kolmogorov-Sinai (K-S) entropy in Cπ. We give examples of ergodic measures in Cπ with zero entropy.

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