Deviation inequalities for separately Lipschitz functionals of iterated random functions

We consider an XX-valued Markov chain X1,X2,…,XnX1,X2,…,Xn belonging to a class of iterated random functions, which is “one-step contracting” with respect to some distance dd on XX. If ff is any separately Lipschitz function with respect to dd, we use a well known decomposition of Sn=f(X1,…,Xn)−E[f(X1,…,Xn)]Sn=f(X1,…,Xn)−E[f(X1,…,Xn)] into a sum of martingale differences dkdk with respect to the natural filtration FkFk. We show that each difference dkdk is bounded by a random variable ηkηk independent of Fk−1Fk−1. Using this very strong property, we obtain a large variety of deviation inequalities for SnSn, which are governed by the distribution of the ηkηk’s. Finally, we give an application of these inequalities to the Wasserstein distance between the empirical measure and the invariant distribution of the chain.