ASYMPTOTIC BEHAVIOR OF ULTIMATELY CONTRACTIVE ITERATED RANDOM LIPSCHITZ FUNCTIONS BY

Abstract. Let (Fn)n0 be a random sequence of i.i.d. global Lip-schitz functions on a complete separable metric space (X, d) with Lipschitz constants L1, L2,... For n 0, denote by Mxn = Fn ◦... ◦ F1(x) and M̂xn = F1 ◦... ◦ Fn(x) the associated sequences of forward and back-ward iterations, respectively. If E log+ L1 < 0 (mean contraction) and E log+ d F1(x0), x0 is finite for some x0 ∈ X, then it is known (see [9]) that, for each x ∈ X, the Markov chain Mxn converges weakly to its unique stationary distribution pi, while M̂xn is a.s. convergent to a random variable M̂ ∞ which does not depend on x and has distribution pi. In [2], renewal theoretic methods have been successfully employed to provide convergence rate results for M̂xn, which then also lead to corresponding assertions fo