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

TL;DR

This paper investigates finitely additive Markov chains on discrete spaces, decomposing their kernels into countably and finitely additive parts, and studies their invariant measures and asymptotic behaviors.

Contribution

It introduces a decomposition of finitely additive Markov kernels into two components and analyzes the properties of their invariant measures and asymptotic regularities.

Findings

Countably additive kernels are atomic.

Invariant measures for purely finitely additive kernels are characterized.

Asymptotic regularities of finitely additive Markov chains are identified.

Abstract

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 $ \Sigma $ is discrete, i.e. is the set of all subsets in $ X $. This construction of the phase space $ (X, \Sigma) $ 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.