New conditions for subgeometric rates of convergence in the Wasserstein distance for Markov chains

Alain Durmus (1) , Eric Moulines (1) , Gersende Fort (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 subgeometric rates of convergence in the Wasserstein distance for general state-space Markov chains which are not phi-irreducible. Our approach is based on a coupling construction adapted to the Wasserstein distance. Our results are applied to establish the subgeometric ergodicity in Wasserstein distance of non-linear autoregressive models in $\mathbb{R}^d$ and of the pre-conditioned Crank-Nicolson algorithm MCMC algorithm in a Hilbert space. In particular, for the latter, we show that a simple Hölder condition on the log-density of the target distribution implies the subgeometric ergodicity of the MCMC sampler in a Wasserstein distance.

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

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

Alain Durmus, Eric Moulines, Gersende Fort. New conditions for subgeometric rates of convergence in the Wasserstein distance for Markov chains. 2014. ⟨hal-00948661v1⟩