Convergence of Markov chain transition probabilities

Michael Scheutzow; Juni Schindler

doi:10.1214/21-ecp395

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arXiv · 2004.10235

Convergence of Markov chain transition probabilities

Michael Scheutzow·Juni Schindler

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Abstract

Consider a discrete time Markov chain with rather general state space which has an invariant probability measure $\mu$. There are several sufficient conditions in the literature which guarantee convergence of all or $\mu$-almost all transition probabilities to $\mu$ in the total variation (TV) metric: irreducibility plus aperiodicity, equivalence properties of transition probabilities, or coupling properties. In this work, we review and improve some of these criteria in such a way that they become necessary and sufficient for TV convergence of all respectively $\mu$-almost all transition probabilities. In addition, we discuss so-called generalized couplings.

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Michael Scheutzow, Juni Schindler. 2020-04-21. Convergence of Markov chain transition probabilities. https://doi.org/10.1214/21-ecp395

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