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Decomposition of Finitely Additive Markov Chains in Discrete Space

Author

Listed:
  • Alexander Zhdanok

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

  • Anna Khuruma

    (Department of Physics and Mathematics, Tuvan State University, 36 Lenina Street, 667000 Kyzyl, Russia)

Abstract

In this study, we consider general Markov chains (MC) defined by a transition probability (kernel) that is finitely additive. These Markov chains were constructed by S. Ramakrishnan within the concepts and symbolism of game theory. Here, we study these MCs by using the operator approach. In our work, the state space (phase space) of the MC has any cardinality and the sigma-algebra is discrete. The construction of a phase space allows us to decompose the Markov kernel (and the Markov operators that it generates) into the sum of two components: countably additive and purely finitely additive kernels. We show that the countably additive kernel is atomic. Some properties of Markov operators with a purely finitely additive kernel and their invariant measures are also studied. A class of combined finitely additive MC and two of its subclasses are introduced, and the properties of their invariant measures are proven. Some asymptotic regularities of such MCs were revealed.

Suggested Citation

  • Alexander Zhdanok & Anna Khuruma, 2022. "Decomposition of Finitely Additive Markov Chains in Discrete Space," Mathematics, MDPI, vol. 10(12), pages 1-21, June.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:12:p:2083-:d:839888
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    References listed on IDEAS

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    1. Sreela Gangopadhyay & B. V. Rao, 1997. "Some Finitely Additive Probability: Random Walks," Journal of Theoretical Probability, Springer, vol. 10(3), pages 643-657, July.
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    Cited by:

    1. Alexander Zhdanok, 2023. "Invariant Finitely Additive Measures for General Markov Chains and the Doeblin Condition," Mathematics, MDPI, vol. 11(15), pages 1-15, August.

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