Deviation inequalities for separately Lipschitz functionals of iterated random functions

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.