Functional central limit theorems for iterated function systems controlled by regenerative sequences

Abstract

Let X be a Polish space and let S be a measurable space. Let {In} be a regenerative process with state space s. Take Z0 arbitrary but independent of {In}. We consider an iterated function system obtained recursively by Zn = FIn-1 (Zn-1)(n ≥ 1), where the function F : X × S → X defined by F(x, s) = Fs (x) is measurable and for each s ∈ S, Fs (x) is a continuous function of x. We obtain sufficient conditions under which, whatever the initial distribution, the functional central limit theorem holds.