Deviation inequalities for separately Lipschitz functionals of iterated random functions

Abstract : We consider a 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. 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 F_{k-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.
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Contributor : Jérôme Dedecker <>
Submitted on : Friday, October 17, 2014 - 7:47:37 PM
Last modification on : Thursday, April 11, 2019 - 4:02:09 PM

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  • HAL Id : hal-00948216, version 2
  • ARXIV : 1402.4105

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Jérôme Dedecker, Xiequan Fan. Deviation inequalities for separately Lipschitz functionals of iterated random functions. Stochastic Processes and their Applications, Elsevier, 2015, 125 (1), pp.60-90. ⟨hal-00948216v2⟩

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