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

Abstract

Assume (Omega , {mathscr {A}}, P) is a probability space, X is a compact metric space with the sigma -algebra {mathscr {B}} of all its Borel subsets and f: X times Omega rightarrow X is {mathscr {B}} otimes {mathscr {A}} -measurable and contractive in mean. We consider the sequence of iterates of f defined on X times Omega ^{{mathbb {N}}} by f^0(x, omega ) = x and f^n(x, omega ) = fbig (f^{n-1}(x, omega ), omega _nbig ) for n in {mathbb {N}}, and its weak limit pi . We show that if psi :X rightarrow {mathbb {R}} is continuous, then for every x in X the sequence left( frac{1}{n}sum _{k=1}^n psi big (f^k(x,cdot )big )right) _{n in {mathbb {N}}} converges almost surely to int _Xpsi dpi . In fact, we are focusing on the case where the metric space is complete and separable.