Markov Chains pp 455-488 | Cite as

Convergence in the Wasserstein Distance

  • Randal DoucEmail author
  • Eric Moulines
  • Pierre Priouret
  • Philippe Soulier
Part of the Springer Series in Operations Research and Financial Engineering book series (ORFE)


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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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Randal Douc
    • 1
    Email author
  • Eric Moulines
    • 2
  • Pierre Priouret
    • 3
  • Philippe Soulier
    • 4
  1. 1.Département CITITelecom SudParisÉvryFrance
  2. 2.Centre de Mathématiques AppliquéesEcole PloytechniquePalaiseauFrance
  3. 3.Université Pierre et Marie CurieParisFrance
  4. 4.Université Paris NanterreNanterreFrance

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