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Invariant Finitely Additive Measures for General Markov Chains and the Doeblin Condition

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  • Alexander Zhdanok

    (Institute for Information Transmission Problems (A.A. Kharkevich Institute) of Russian Academy of Sciences, 19/1 Bolshoy Karetny Per., 127051 Moscow, Russia)

Abstract

In this paper, we consider general Markov chains with discrete time in an arbitrary measurable (phase) space. Markov chains are given by a classical transition function that generates a pair of conjugate linear Markov operators in a Banach space of measurable bounded functions and in a Banach space of bounded finitely additive measures. We study sequences of Cesaro means of powers of Markov operators on the set of finitely additive probability measures. It is proved that the set of all limit measures (points) of such sequences in the weak topology generated by the preconjugate space is non-empty, weakly compact, and all of them are invariant for this operator. We also show that the well-known Doeblin condition ( D ) for the ergodicity of a Markov chain is equivalent to condition ( ∗ ) : all invariant finitely additive measures of the Markov chain are countably additive, i.e., there are no invariant purely finitely additive measures. We give all the proofs for the most general case.

Suggested Citation

  • Alexander Zhdanok, 2023. "Invariant Finitely Additive Measures for General Markov Chains and the Doeblin Condition," Mathematics, MDPI, vol. 11(15), pages 1-15, August.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:15:p:3388-:d:1209363
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    References listed on IDEAS

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    1. Alexander Zhdanok & Anna Khuruma, 2022. "Decomposition of Finitely Additive Markov Chains in Discrete Space," Mathematics, MDPI, vol. 10(12), pages 1-21, June.
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