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Pré-Publication, Document De Travail Année : 2014

McDiarmid's martingale for a class of iterated random functions

Résumé

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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Dates et versions

hal-00948216 , version 1 (17-02-2014)
hal-00948216 , version 2 (17-10-2014)

Identifiants

Citer

Jérôme Dedecker, Xiequan Fan. McDiarmid's martingale for a class of iterated random functions. 2014. ⟨hal-00948216v1⟩

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