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Alexander Zhdanok

Publications and source records attributed to Alexander Zhdanok.

2 recordsLinked to original sources
arXiv2023

Weak ergodic theorem for Markov chains without invariant countably additive measures

In this paper, we study Markov chains (MC) on topological spaces within the framework of the operator approach. We extend the Markov operator from the space of countably additive measures to the space of finitely additive measures. Cesaro means for a Markov sequence of measures and their asymptotic behavior in the weak topology are considered. It is proved ergodic theorem that in order for the Cesaro means to converge weakly to some bounded regular finitely additive (or countably additive) measure it is necessary and sufficient that all invariant finitely additive measures are not separable from the limit measure in the weak topology. Moreover, the limit measure may not be invariant for a MC, and may not be countably additive. The corresponding example is given and studied in detail.

math.PR↗
arXiv2022

Decompositions of finitely additive Markov chains and invariant measures in discrete space

In this paper, we consider general Markov chains (MC), specified by the transition probability (kernel) $ P (x, E) $, finitely additive in the second argument. Such MC are studied within the framework of the functional operator treatment. The state space (phase space) of the MC $ X $ has any cardinality, and the sigma-algebra $ Σ$ is discrete, i.e. is the set of all subsets in $ X $. This construction of the phase space $ (X, Σ) $ allows us to decompose the Markov kernel $ P (x, E) $ into the sum of two components - countably additive and purely finitely additive in the second argument and measurable in the first argument. It is shown that the countably additive kernel is atomic. Some properties of Markov operators with a purely finitely additive kernel and their invariant measures are studied. A class of combined finitely additive MC and two of its subclasses are introduced, and some properties of their invariant measures are proved. Some asymptotic regularities of such MC are revealed.

math.PR↗