arXiv subjects

Alexander I. Zhdanok

Publications and source records attributed to Alexander I. Zhdanok.

3 recordsLinked to original sources

arXiv2020

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

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

math.PR↗

arXiv2020

Ergodicity conditions for general Markov chains in terms of invariant finitely additive measures

We consider general Markov chains with discrete time in an arbitrary measurable (phase) space and homogeneous in time. Markov chains are defined by the classical transition function which within the framework of the operator treatment generates a conjugate pair of linear Markov operators in the Banach space of measurable bounded functions and in the Banach space of bounded finite additive measures. It is proved that the well-known Doeblin condition $ (D) $ of ergodicity (quasi\-compactness) of the Markov chain is equivalent to the condition $ (*) $: all finitely additive invariant measures of the Markov operator are countably additive i.e. there are no invariant purely finitely additive measures. Under some assumptions, it is proved that the conditions $ (D) $ and $ (*) $ are also equivalent to the condition $ (**) $: the set of invariant finitely additive measures of a Markov operator is finite-dimensional. Ergodic theorems are given.

math.PR↗

arXiv2018

Dimension of the space of invariant finitely additive measures of general Markov chains and their ergodic properties

General Markov chains with a countably additive transition probability in arbitrary phase space are considered. Markov operators extend from the space of countably additive measures to the space of finitely additive measures. In the author's papers a theorem was earlier proved that if all invariant finitely additive measures of a Markov chain are countably additive, i.e. there are no invariant purely finitely additive measures, then their subspace is finite-dimensional and the Markov chain satisfies the Doob-Doeblin quasicompactness conditions. In the same paper, a partial inversion of this theorem was proved with the dimension "one". In this paper we prove the inversion of this assertion for any finite dimensionality, but under certain additional conditions. The ergodic consequences are given. Examples and methods for studying their asymptotics with the aid of invariant purely finitely additive measures are given.

math.PR↗