Weak ergodic theorem for Markov chains without invariant countably additive measures

TL;DR

This paper investigates the asymptotic behavior of Markov chains on topological spaces by extending the operator approach to finitely additive measures, establishing conditions for weak convergence of Cesaro means without requiring invariant countably additive measures.

Contribution

It introduces a new ergodic theorem for Markov chains that do not necessarily have invariant countably additive measures, broadening the understanding of their long-term behavior.

Findings

Cesaro means converge weakly under specific conditions.

Invariant finitely additive measures influence convergence.

Limit measures may be non-invariant and not countably additive.

Abstract

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.