General Markov Chains: Cycles of Finitely Additive Measures and\n Classical Cycles of States

2020/12/08 by Alexander I. Zhdanok, Zhdanok, Alexander I.
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #28A33 #46E27 #60F05 #60J05 #FOS: Mathematics #Functional Analysis (math.FA) #Gene Regulatory Network Analysis #Mathematical Control Systems and Analysis #Probability (math.PR) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2012.04547

openalex publication_date 2020/12/08 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

General Markov chains in an arbitrary phase space are considered in the\nframework of the operator treatment. Markov operators continue from the space\nof countably additive measures to the space of finitely additive measures.\nCycles of measures generated by the corresponding operator are constructed, and\nalgebraic operations on them are introduced. One of the main results obtained\nis that any cycle of finitely additive measures can be uniquely decomposed into\nthe coordinate-wise sum of a cycle of countably additive measures and a cycle\nof purely finitely additive measures.We have proved theorems on the conditions\nand consequences of consistency cycles of measures with cycles of sets of\nstates of General Markov chains. A theorem is proved (under certain conditions)\nthat if a finitely additive cycle of a Markov chain is unique, then it is\ncountably additive.\n

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