Limit theorems for iterated random functions by regenerative methods

Abstract

Let ( X,d) be a complete separable metric space and ( F n ) n⩾0 a sequence of i.i.d. random functions from X to X which are uniform Lipschitz, that is, L n= sup x≠y d(F n(x),F n(y))/d(x,y)<∞ a.s. Providing the mean contraction assumption E log + L 1<0 and E log + d(F 1(x 0),x 0)<∞ for some x 0∈ X , it was proved by Elton (Stochast. Proc. Appl. 34 (1990) 39–47) that the forward iterations M n x = F n ∘⋯∘ F 1( x), n⩾0, converge weakly to a unique stationary distribution π for each x∈ X . The associated backward iterations M ̂ n x=F 1∘⋯∘F n(x) are a.s. convergent to a random variable M ̂ ∞ which does not depend on x and has distribution π. Based on the inequality d( M ̂ n+m x, M ̂ n x)⩽ exp(∑ k=1 n log L k)d(F n+1∘⋯∘F n+m(x),x) for all n, m⩾0 and the observation that ( ∑ k=1 n log L k) n⩾0 forms an ordinary random walk with negative drift, we will provide new estimates for d( M ̂ ∞, M ̂ n x) and d( M n x , M n y ), x,y∈ X , under polynomial as well as exponential moment conditions on log(1+ L 1) and log(1+ d( F 1( x 0), x 0)). It will particularly be shown, that the decrease of the Prokhorov distance between P n ( x,·) and π to 0 is of polynomial, respectively exponential rate under these conditions where P n 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.