-continuous Markov chains. II

Author:
Shu-teh C. Moy

Journal:
Trans. Amer. Math. Soc. **120** (1965), 83-107

MSC:
Primary 60.65

DOI:
https://doi.org/10.1090/S0002-9947-1965-0183020-8

MathSciNet review:
0183020

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Abstract: Continuing the investigation in [8] we study a -continuous Markov operator . It is shown that, if is conservative and ergodic, is indeed ``periodic'' as is the case when the state space is discrete; there is a positive integer , called the period of , such that the state space may be decomposed into cyclically moving sets and, for every positive integer acting on each alone is ergodic. It is also shown that maps into where is the nontrivial invariant measure of and . If is finite and normalized then it is shown that (1) if , then converges a.e. to where if and if , (2) converges in to if , and(3) a.e. if and . If is infinite, then it is shown that (1) if for some , then a.e. , (2) there exists a sequence of sets such that and a.e. for and .

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DOI:
https://doi.org/10.1090/S0002-9947-1965-0183020-8

Article copyright:
© Copyright 1965
American Mathematical Society