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

TL;DR

This paper explores the structure of cycles in general Markov chains within arbitrary phase spaces, extending the operator framework to finitely additive measures and analyzing their decomposition and relation to state cycles.

Contribution

It introduces a novel decomposition of finitely additive measure cycles into countably additive and purely finitely additive parts, and establishes conditions linking measure cycles to state cycles.

Findings

Any finitely additive measure cycle can be uniquely decomposed into countably additive and purely finitely additive cycles.

Under certain conditions, a unique finitely additive cycle must be countably additive.

Theorems relate measure cycles to cycles of sets of states in general Markov chains.

Abstract

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.