Deviation inequalities for separately Lipschitz functionals of iterated random functions

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:125

Article

Dedecker, Jerome^1,2  Fan, Xiequan^3,4 

[1] Univ Paris 05, Sorbonne Paris Cite, Lab MAP5, F-75016 Paris, France

[2] CNRS, UMR 8145, F-75016 Paris, France

[3] Inria, Regular Team, F-92295 Chatenay Malabry, France

[4] Ecole Cent Paris, MAS Lab, F-92295 Chatenay Malabry, France

关键词: Iterated random functions;    Martingales;    Exponential inequalities;    Moment inequalities;    Wasserstein distances;   

DOI  :  10.1016/j.spa.2014.08.001

来源: Elsevier

【 摘 要 】

We consider an X-valued Markov chain X-1, X-2, ..., X-n belonging to a class of iterated random functions, which is one-step contracting with respect to some distance d on chi. If f is any separately Lipschitz function with respect to d, we use a well known decomposition of S-n = f(X-1, ..., X-n) - E[f(X-1, ..., X-n)] into a sum of martingale differences d(k) with respect to the natural filtration F-k. We show that each difference d(k) is bounded by a random variable eta(k) independent of Fk-1. Using this very strong property, we obtain a large variety of deviation inequalities for S-n, which are governed by the distribution of the eta(k)'s. Finally, we give an application of these inequalities to the Wasserstein distance between the empirical measure and the invariant distribution of the chain. (C) 2014 Elsevier B.V. All rights reserved.

【 授权许可】

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