Subgeometric rates of convergence in Wasserstein distance for Markov chains

Alain Durmus; Gersende Fort; Éric Moulines

doi:10.1214/15-AIHP699

Subgeometric rates of convergence in Wasserstein distance for Markov chains

Alain Durmus, Gersende Fort, Éric Moulines

CNRS LTCI

Research output: Contribution to journal › Article › peer-review

Abstract

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 (Ann. Appl. Probab. 24 (2) (2014) 526-552), 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 in (Bernoulli 13 (3) (2007) 831-848). Our results are applied to establish the subgeometric ergodicity inWasserstein distance of non-linear autoregressive models and of the pre-conditioned Crank-Nicolson Markov chain Monte Carlo algorithm in Hilbert space.

Original language	English
Pages (from-to)	1799-1822
Number of pages	24
Journal	Annales de l'institut Henri Poincare (B) Probability and Statistics
Volume	52
Issue number	4
DOIs	https://doi.org/10.1214/15-AIHP699
Publication status	Published - 1 Nov 2016
Externally published	Yes

Keywords

Markov chain Monte Carlo in infinite dimension
Markov chains
Subgeometric ergodicity
Wasserstein distance

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10.1214/15-AIHP699

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title = "Subgeometric rates of convergence in Wasserstein distance for Markov chains",

abstract = "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 (Ann. Appl. Probab. 24 (2) (2014) 526-552), 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 in (Bernoulli 13 (3) (2007) 831-848). Our results are applied to establish the subgeometric ergodicity inWasserstein distance of non-linear autoregressive models and of the pre-conditioned Crank-Nicolson Markov chain Monte Carlo algorithm in Hilbert space.",

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T1 - Subgeometric rates of convergence in Wasserstein distance for Markov chains

AU - Durmus, Alain

AU - Fort, Gersende

AU - Moulines, Éric

N1 - Publisher Copyright: © Association des Publications de l'Institut Henri Poincaré, 2016.

PY - 2016/11/1

Y1 - 2016/11/1

N2 - 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 (Ann. Appl. Probab. 24 (2) (2014) 526-552), 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 in (Bernoulli 13 (3) (2007) 831-848). Our results are applied to establish the subgeometric ergodicity inWasserstein distance of non-linear autoregressive models and of the pre-conditioned Crank-Nicolson Markov chain Monte Carlo algorithm in Hilbert space.

AB - 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 (Ann. Appl. Probab. 24 (2) (2014) 526-552), 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 in (Bernoulli 13 (3) (2007) 831-848). Our results are applied to establish the subgeometric ergodicity inWasserstein distance of non-linear autoregressive models and of the pre-conditioned Crank-Nicolson Markov chain Monte Carlo algorithm in Hilbert space.

KW - Markov chain Monte Carlo in infinite dimension

KW - Markov chains

KW - Subgeometric ergodicity

KW - Wasserstein distance

U2 - 10.1214/15-AIHP699

DO - 10.1214/15-AIHP699

M3 - Article

AN - SCOPUS:84997181555

SN - 0246-0203

VL - 52

SP - 1799

EP - 1822

JO - Annales de l'institut Henri Poincare (B) Probability and Statistics

JF - Annales de l'institut Henri Poincare (B) Probability and Statistics

IS - 4

ER -

Subgeometric rates of convergence in Wasserstein distance for Markov chains

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Keywords

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