Consider a discrete time Markov chain with rather general state space which has an invariant probability measure µ. There are several sufficient conditions in the literature which guarantee convergence of all or µ-almost all transition probabilities to µ 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 µ-almost all transition probabilities.In addition, we discuss so-called generalized couplings.
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