General Markov Chains: Dimension of the Space of Invariant Finitely Additive Measures and Their Ergodicity—Problematic Examples

Alexander Zhdanok ()
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Alexander Zhdanok: Institute for Information Transmission Problems (A.A. Kharkevich Institute), Russian Academy of Sciences, Bolshoy Karetny Per. 19, Building 1, 127051 Moscow, Russia

Mathematics, 2025, vol. 13, issue 22, 1-22

Abstract: This study considers general Markov chains (MCs) with discrete time in an arbitrary phase space. The transition function of the MC generates two operators: T , which acts on the space of measurable functions, and A , which acts on the space of bounded countably additive measures. The operator T * , which is adjoint to T and acts on the space of finitely additive measures, is also constructed. A number of theorems are proved for the operator T * , including the ergodic theorem. Under certain conditions it is proved that if the MC has a finite number of basic invariant finitely additive measures then all of them are countably additive and the MC is quasi-compact. We demonstrate a methodology that applies finitely additive measures for the analysis of MCs, using examples with detailed proofs of their non-simple properties. Some of these proofs in the examples are more complicated than the proofs in our theorems.

Keywords: general markov chains; markov operators; finitely additive measures; invariant measures; quasi-compactness; ergodic theorem (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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