2021/05/21 by Alexander I. Zhdanok, Zhdanok, Alexander, Anna Khuruma +1
Computer Science · Mathematics · #46E27 #60J05 #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Equations Stability Results #Matrix Theory and Algorithms #Probability (math.PR) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2105.10418
openalex publication_date 2021/05/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we consider general Markov chains (MC), specified by the\ntransition probability (kernel) P (x, E) , finitely additive in the second\nargument. Such MC are studied within the framework of the functional operator\ntreatment. The state space (phase space) of the MC X has any cardinality,\nand the sigma-algebra \Σ is discrete, i.e. is the set of all subsets in\n X . This construction of the phase space (X, \Σ) allows us to\ndecompose the Markov kernel P (x, E) into the sum of two components -\ncountably additive and purely finitely additive in the second argument and\nmeasurable in the first argument. It is shown that the countably additive\nkernel is atomic. Some properties of Markov operators with a purely finitely\nadditive kernel and their invariant measures are studied. A class of combined\nfinitely additive MC and two of its subclasses are introduced, and some\nproperties of their invariant measures are proved. Some asymptotic regularities\nof such MC are revealed.\n