Limit theorems for iterated random functions by regenerative methods

Let (X,d) be a complete separable metric space and (Fn)n⩾0 a sequence of i.i.d. random functions from X to X which are uniform Lipschitz, that is, Ln=supx≠y d(Fn(x),Fn(y))/d(x,y)<∞ a.s. Providing the mean contraction assumption E log+ L1<0 and E log+ d(F1(x0),x0)<∞ for some x0∈X, it was proved by Elton (Stochast. Proc. Appl. 34 (1990) 39–47) that the forward iterations Mnx=Fn∘⋯∘F1(x), n⩾0, converge weakly to a unique stationary distribution π for each x∈X. The associated backward iterations M̂nx=F1∘⋯∘Fn(x) are a.s. convergent to a random variable M̂∞ which does not depend on x and has distribution π. Based on the inequality d(M̂n+mx,M̂nx)⩽exp(∑k=1n log Lk)d(Fn+1∘⋯∘Fn+m(x),x) for all n,m⩾0 and the observation that (∑k=1n log Lk)n⩾0 forms an ordinary random walk with negative drift, we will provide new estimates for d(M̂∞,M̂nx) and d(Mnx,Mny), x,y∈X, 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 π 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.