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

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