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Asymptotic behavior of ultimately contractive iterated random Lipschitz functions

Alsmeyer G., Hölker G.

Probability and Mathematical Statistics

2009

Vol. 29, Fasc. 2

321--336

Let (Fn)n≥0 be a random sequence of i.i.d. global Lipschitz functions on a complete separable metric space (X; d) with Lipschit constants L1; L2; : : : For n ≥0, denote by Mx n = Fn○ : : : ○ F1(x) and ^Mx n = Fn○ : : : ○ F1(x) the associated sequences of forward and backward iterations, respectively. If E log+ L1 < 0 (mean contraction) and E log+ d ( F1(x0); x0) is finite for some x0ЄX, then it is known (see [9]) that, for each x Є X, the Markov chain Mx n converges weakly to its unique stationary distribution π, while ^M xn is a.s. convergent to a random variable ^M∞ which does not depend on x and has distribution π. In [2], renewal theoretic methods have been successfully employed to provide convergence rate results for ^M x n, which then also lead to corresponding assertions for Mx n via Mx n d= ^M x n for all n and x, where d= means equality in law. Here our purpose is to demonstrate how these methods are extended to the more general situation where only ultimate contraction, i.e. an a.s. negative Lyapunov exponent limn→∞ n−1 log l(Fn○ : : : ○ F1) is assumed (here l(F) denotes the Lipschitz constant of F). This not only leads to an extension of the results from [2] but in fact also to improvements of the obtained convergence rate.