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 Mat. Tr., 2002, Volume 5, Number 1, Pages 46–65 (Mi mt98)

Finitely Additive Measures in the Ergodic Theory of Markov Chains. II

A. I. Zhdanok

Tyva State University

Abstract: We develop a new approach to the study of general Markov chains (MC), i.e., homogeneous Markov processes with discrete time on an arbitrary phase space. In the first part of the article, we suggested an extension of the traditional space of countably additive measures to the space of finitely additive measures. Given an arbitrary phase space, we constructed its “gamma-compactification” to which we extended each Markov chain. We established an isomorphism between all finitely additive Markov chains on the initial space and Feller countably additive chains on its “gamma-compactification”. Using the above construction, in the second part, we prove weak and strong ergodic theorems that establish a substantial dependence of the asymptotic behavior of a Markov chain on the presence and properties of invariant finitely additive measures. The study in the article is carried out in the framework of functional operator approach.

Key words: finitely additive measure, countably additive measure, Markov chain, Markov operators, arbitrary phase space, compactification of an arbitrary phase space, extension of a Markov chain to the compactification, invariant measure, ergodic theorems.

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English version:
Siberian Advances in Mathematics, 2003, 13:2, 108–125

Bibliographic databases:

UDC: 519.21+517.98+515.12

Citation: A. I. Zhdanok, “Finitely Additive Measures in the Ergodic Theory of Markov Chains. II”, Mat. Tr., 5:1 (2002), 46–65; Siberian Adv. Math., 13:2 (2003), 108–125

Citation in format AMSBIB
\Bibitem{Zhd02} \by A.~I.~Zhdanok \paper Finitely Additive Measures in the~Ergodic Theory of Markov Chains.~II \jour Mat. Tr. \yr 2002 \vol 5 \issue 1 \pages 46--65 \mathnet{http://mi.mathnet.ru/mt98} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1918894} \zmath{https://zbmath.org/?q=an:1046.60069|1036.60064} \transl \jour Siberian Adv. Math. \yr 2003 \vol 13 \issue 2 \pages 108--125 

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This publication is cited in the following articles:
1. A. I. Zhdanok, “Gamma-compactification of measurable spaces”, Siberian Math. J., 44:3 (2003), 463–476
2. A. E. Gutman, A. I. Sotnikov, “Order properties of the space of finitely additive transition functions”, Siberian Math. J., 45:1 (2004), 69–85
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