Markov Chains pp 455-488 | Cite as

# Convergence in the Wasserstein Distance

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## Abstract

In the previous chapters, we obtained rates of convergence in the total variation distance of the iterates \(P^n\) of an irreducible positive Markov kernel *P* to its unique invariant measure \(\pi \) for \(\pi \)-almost every \(x \in \mathsf {X}\) and for all \(x \in \mathsf {X}\) if the kernel *P* is irreducible and positive Harris recurrent. Conversely, convergence in the total variation distance for all \(x\in \mathsf {X}\) entails irreducibility and that \(\pi \) be a maximal irreducibility measure.

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