Subgeometric rates of convergence in Wasserstein distance for Markov chains

Alain Durmus (1) , Gersende Fort (1) , Éric Moulines (1)

1 LTCI - Laboratoire Traitement et Communication de l'Information

In this paper, we provide sufficient conditions for the existence of the invariant distribution and for subgeometric rates of convergence in Wasserstein distance for general state-space Markov chains which are (possibly) not irreducible. Compared to previous work, our approach is based on a purely probabilistic coupling construction which allows to retrieve rates of convergence matching those previously reported for convergence in total variation. Our results are applied to establish the subgeometric ergodicity in Wasserstein distance of non-linear autoregressive models and of the pre-conditioned Crank-Nicolson Markov chain Monte Carlo algorithm in Hilbert space.

Mots clés: Wasserstein distance; rate of convergence; Markov chain Monte Carlo in infinite dimension; Markov chains

Domaines: Probabilités [math.PR]; Statistiques [math.ST]; Théorie [stat.TH]

Alain Durmus, Gersende Fort, Éric Moulines. Subgeometric rates of convergence in Wasserstein distance for Markov chains. 2014. ⟨hal-00948661v2⟩