Empirical processes of iterated maps that contract on average

We consider a Markov chain obtained by random iterations of Lipschitz maps Ti chosen with a probability pi(x) depending on the current position x. We assume this system has a property of “contraction on average”, that is ∑id(Tix,Tiy)pi(x)<ρd(x,y) for some ρ<1. In the present note, we study the weak convergence of the empirical process associated to this Markov chain.