Limit theorems for iterated random functions by regenerative methods

Let (X,d) be a complete separable metric space and (F-n)(n greater than or equal to0) a sequence of i.i.d. random functions from X to X which are unif orm Lipschitz, that is, L-n=sup(x not equaly)d(F-n(x),F-n(y))/d(x,y) < infi nity a.s. Providing the mean contraction assumption Elog(+)L(1) < 0 and Elo g(+)d(F-1(x(0)),x(0)) < infinity for some x(0) is an element of X, it was p roved by Elton (Stochast. Proc. Appl. 34 (1990) 39-47) that the forward ite rations M-n(x) = F-n o...o F-1(x), n greater than or equal to 0, converge w eakly to a unique stationary distribution pi for each x is an element of X. The associated backward iterations (M) over cap (x)(n) = F-1 o...o F-n(x) are a.s. convergent to a random variable (M) over cap infinity which does n ot depend on x and has distribution pi. Based on the inequality d((M) over cap (x)(n+m), (M) over cap (x)(n)) less than or equal to exp(Sigma (n)(k=1) logL(k))d(Fn+1 o...o Fn+m(x),x) for all n,m greater than or equal to 0 and the observation that (F-k=1(n) logL(k))(n greater than or equal to0) forms an ordinary random walk with negative drift, we will provide new estimates for d((M) over cap (infinity),(M) over cap (x)(n)) and d(M-n(x),M-n(y)), x , y is an element of X, under polynomial as well as exponential moment cond itions on log(1 + L-1) and log(1 + d(F-1(x(0)), x(0))). It will particularl y be shown, that the decrease of the Prokhorov distance between P-n(x, (.)) and pi to 0 is of polynomial, respectively exponential rate under these co nditions where P-n denotes the n-step transition kernel of the Markov chain of forward iterations. The exponential rate was recently proved by Diaconi s and Freedman (SIAM Rev. 41 (1999) 45-76) using different methods. (C) 200 1 Elsevier Science B.V. All rights reserved.