Strong Law of Large Numbers for Iterates of Some Random-Valued Functions

Abstract

Assume (Ω,A,P) is a probability space, X is a compact metric space with the σ-algebra B of all its Borel subsets and f:X×Ω→X is B⊗A-measurable and contractive in mean. We consider the sequence of iterates of f defined on X×ΩN by f0(x,ω)=x and fn(x,ω)=f(fn−1(x,ω),ωn) for n∈N, and its weak limit π. We show that if ψ:X→R is continuous, then for every x∈X the sequence (1n∑nk=1ψ(fk(x,⋅)))n∈N converges almost surely to ∫Xψdπ. In fact, we are focusing on the case where the metric space is complete and separable.