Weak ergodic theorem for Markov chains in the absence of invariant countably additive measures

Abstract

General Markov chains (MC) with a countably additive transition probability on a normal topological space are considered. We extend the Markov operator from the traditional space of countably additive measures to the space of finitely additive measures. We study the Cesaro means for the Markov sequence of measures and their asymptotic behavior in the weak topology generated by the space of bounded continuous functions. 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 (such always exist) 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. Key words: Markov chain, Markov operators, Cesaro means, weak ergodic theorem, absence of invariant countably additive measures, invariant finitely additive measure, purely finitely additive measures.Comment: 15 pages, 1 figur