Convergence of Markov chain transition probabilities

Abstract

Consider a discrete time Markov chain with rather general state space which has an invariant probability measure $\hat{I}¼$. There are several sufficient conditions in the literature which guarantee convergence of all or $\hat{I}¼$-almost all transition probabilities to $\hat{I}¼$ 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 $\hat{I}¼$-almost all transition probabilities. In addition, we discuss so-called generalized couplings.