Weak law of large numbers for iterates of random-valued functions

Abstract

Given a probability space (Ω,A, P), a complete and separable metric space X with the σ-algebra B of all its Borel subsets and a B⊗A-measurable f : X ×Ω → X we consider its iterates fn defined on X × ΩN by f0(x, ω) = x and fn(x, ω) = f fn−1(x, ω), ωn for n ∈ N and provide a simple criterion for the existence of a probability Borel measure π on X such that for every x ∈ X and for every Lipschitz and bounded ψ : X → R the sequence 1 n n−1 k=0 ψ fk(x, ·) n∈N converges in probability to X ψ(y)π(dy).