Deviation inequalities for separately Lipschitz functionals of iterated random functions

We consider an X-valued Markov chain X1,X2,…,Xn belonging to a class of iterated random functions, which is “one-step contracting” with respect to some distance d on X. If f is any separately Lipschitz function with respect to d, we use a well known decomposition of Sn=f(X1,…,Xn)−E[f(X1,…,Xn)] into a sum of martingale differences dk with respect to the natural filtration Fk. We show that each difference dk is bounded by a random variable ηk independent of Fk−1. Using this very strong property, we obtain a large variety of deviation inequalities for Sn, which are governed by the distribution of the ηk’s. Finally, we give an application of these inequalities to the Wasserstein distance between the empirical measure and the invariant distribution of the chain.