Empirical processes of iterated maps that contract on average

We consider a Markov chain obtained by random iterations of Lipschitz maps $T_i$ chosen with a probability $p_i(x)$ depending on the current position $x$. We assume this system has a property of "contraction on average", that is $\sum_i d(T_ix,T_iy)p_i(x) < ρd(x,y)$ for some $ρ<1$. In the present note, we study the weak convergence of the empirical process associated to this Markov chain.

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