Limit theorems for iterated random functions by regenerative methods

Let be a complete separable metric space and (Fn)n[greater-or-equal, slanted]0 a sequence of i.i.d. random functions from to which are uniform Lipschitz, that is, Ln=supx[not equal to]y d(Fn(x),Fn(y))/d(x,y)<[infinity] a.s. Providing the mean contraction assumption and for some , it was proved by Elton (Stochast. Proc. Appl. 34 (1990) 39-47) that the forward iterations Mnx=Fno...oF1(x), n[greater-or-equal, slanted]0, converge weakly to a unique stationary distribution [pi] for each . The associated backward iterations are a.s. convergent to a random variable which does not depend on x and has distribution [pi]. Based on the inequality for all n,m[greater-or-equal, slanted]0 and the observation that forms an ordinary random walk with negative drift, we will provide new estimates for and d(Mnx,Mny), , under polynomial as well as exponential moment conditions on log(1+L1) and log(1+d(F1(x0),x0)). It will particularly be shown, that the decrease of the Prokhorov distance between Pn(x,·) and [pi] to 0 is of polynomial, respectively exponential rate under these conditions where Pn denotes the n-step transition kernel of the Markov chain of forward iterations. The exponential rate was recently proved by Diaconis and Freedman (SIAM Rev. 41 (1999) 45-76) using different methods.