Decomposition of Finitely Additive Markov Chains in Discrete Space

Mathematics

Anna Khuruma¹  Alexander Zhdanok² 

[1] Department of Physics and Mathematics, Tuvan State University, 36 Lenina Street, 667000 Kyzyl, Russia;Institute for Information Transmission Problems of the Russian Academy of Sciences, 19, build.1, Bolshoy Karetny Per., 127051 Moscow, Russia;

关键词: Markov chains;    Markov operators;    finitely additive Markov chains;    finitely additive measures;    invariant measures;    decompositions of Markov chains;   

DOI  :  10.3390/math10122083

来源: DOAJ

【 摘 要 】

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.

【 授权许可】

Unknown