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

Assume (Ω,A,P)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ (\Omega , {\mathscr {A}}, P) $$\end{document} is a probability space, X is a compact metric space with the σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sigma $$\end{document}-algebra B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathscr {B}} $$\end{document} of all its Borel subsets and f:X×Ω→X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ f: X \times \Omega \rightarrow X $$\end{document} is B⊗A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathscr {B}} \otimes {\mathscr {A}} $$\end{document}-measurable and contractive in mean. We consider the sequence of iterates of f defined on X×ΩN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ X \times \Omega ^{{\mathbb {N}}}$$\end{document} by f0(x,ω)=x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f^0(x, \omega ) = x$$\end{document} and fn(x,ω)=f(fn-1(x,ω),ωn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ f^n(x, \omega ) = f\big (f^{n-1}(x, \omega ), \omega _n\big )$$\end{document} for n∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \in {\mathbb {N}}$$\end{document}, and its weak limit π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi $$\end{document}. We show that if ψ:X→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi :X \rightarrow {\mathbb {R}}$$\end{document} is continuous, then for every x∈X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ x \in X $$\end{document} the sequence 1n∑k=1nψ(fk(x,·))n∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( \frac{1}{n}\sum _{k=1}^n \psi \big (f^k(x,\cdot )\big )\right) _{n \in {\mathbb {N}}}$$\end{document} converges almost surely to ∫Xψdπ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int _X\psi d\pi $$\end{document}. In fact, we are focusing on the case where the metric space is complete and separable.


Introduction
Fix a probability space (Ω, A , P ) and a metric space X.
Let B denote the σ-algebra of all Borel subsets of X. We say that f : X × Ω → X is a random-valued function (shortly: an rv-function) if it is measurable with respect to the product σ-algebra B ⊗ A . The iterates of such an rv-function are given by with A from the product σ-algebra A n . See [10,Sec. 1.4], [8].
A result on a.s. convergence of f n (x, ·) n∈N for X being the unit interval can be found in [10,Sec. 1.4B]. The paper [7] brings theorems on the convergence a.s. and in L 1 of those sequences of iterates in the case where X is a closed subset of a separable Banach lattice. A simple criterion for the convergence in law of f n (x, ·) n∈N to a random variable independent of x ∈ X was proved in [1], assuming that X is complete and separable. In [2] it has been strengthened and applied to obtain a weak law of large numbers for iterates of random-valued functions. In the present paper we are interested in a strong law of large numbers. We will be based on the following Brunk-Prokhorov-type theorem, see [11,Theorem 3.3.1] and [6, Corollary 3.1].
(C) Let (F n ) n∈N be an increasing sequence of sub-σ-algebras of A and (ξ n ) n∈N a sequence of random variables such that ξ n is F n -measurable and E(ξ n+1 |F n ) = 0 for each n ∈ N. If (a n ) n∈N is an increasing and unbounded sequence of positive reals and then lim n→∞ 1 a n n k=1 ξ k = 0 a.s.

A Scheme
Assume X is a metric space and f : X × Ω → X an rv-function.
x ∈ X and n ∈ N, then the function α : X → R defined by is Borel and Consequently, for every x ∈ X and n ∈ N the function α • f n (x, ·) is A nmeasurable and for each A ∈ A n we have The following theorem is in fact a scheme of proving a strong law of large numbers for iterates of random-valued functions. Proposition 1. Let ψ : X → R and assume that there exists a Borel and bounded ϕ : X → R such that If (a n ) n∈N is an increasing and unbounded sequence of positive reals such that Proof. Define α : Then |ξ n | ≤ 2M and by Lemma 1, E(ξ n+1 |A n ) = 0 for each n ∈ N. It now follows from Brunk-Prokhorov-type theorem (C) that for every n ∈ N. Moreover, |α • f n (x, ·)| ≤ M . Consequently (3) holds.

The Weak Limit
Assume now the following hypothesis (H).
(H) (X, ρ) is a complete and separable metric space and f : with a λ ∈ (0, 1), and Then (see [1, Theorem 3.1]) there exists a probability Borel measure π f on X such that for every x ∈ X the sequence of distributions of f n (x, ·), n ∈ N, converges weakly to π f . See also [3, Lemma 2.2] and [9, Corollary 5.6 and Lemma 3.1].
This limit distribution π f plays an important role in solving functional equations, in particular in the class of Hölder continuous functions. We call a function ψ : Moreover we call a function Hölder continuous if it is Hölder continuous with an exponent δ ∈ (0, 1]. The following theorem (see [3, Theorem 2.1] and [4, Corollary 2.6]) will be useful to us.
(B) Assume (H). If ψ : X → R is Hölder continuous with exponent δ ∈ (0, 1], then it is integrable for π f and if additionally then there exists a Hölder continuous with exponent δ function ϕ : X → R such that (2) holds.

Main Results
In what follows (X, ρ) is a metric space and f : X × Ω → X is an rv-function. We start with a simple consequence of Proposition 1 and (B). It is a special case of Theorem 2 given below, but shows our approach without technical details. Proof. Fix a Hölder continuous ψ : X → R. Replacing ψ by ψ − X ψdπ f we may assume that (9) holds. By (B) there is a Hölder continuous ϕ : X → R satisfying (2). Since X is bounded, so is ϕ. Applying now Proposition 1 with a n = n for n ∈ N we obtain (3) which ends the proof.
Since continuous real functions defined on a compact metric space can be uniformly approximated by Lipschitz functions (see [5, 11.2.4]), Theorem 1 implies the following corollary. (7) holds with a λ ∈ (0, 1), then we have (10) for every continuous ψ : X → R and for each x ∈ X.

Lemma 2.
For every x ∈ X and n ∈ N we have Proof. Fix x ∈ X, n ∈ N and assume for the inductive proof that Then, applying Fubini's theorem, (7) and the above inequality, we obtain which ends the proof of the first part. To get the second one observe that by (12) and Jensen's inequality for every x ∈ X and n ∈ N we have Lemma 2 makes sense to define a Borel function α : X → R by (1).

Lemma 3.
For every x ∈ X and n ∈ N we have Proof. Since, for every ω ∈ Ω ∞ and ω ∈ Ω, Hence, applying Jensen's inequality and Fubini's theorem,

Lemma 4.
Let (b n ) n∈N be a converging to zero sequence of positive reals. If Proof. If n ∈ N and ω ∈ Ω, then by (1), (12), Jensen's inequality and (7) we have Now to finish the proof it is enough to show that lim n→∞ b n ξ n = 0 a.e. for P ∞ , where ξ n = ρ f n (x, ·), x δ for n ∈ N. To this end observe that by Markov's inequality for every n ∈ N and ε > 0 we have Hence it follows from the assumption of the lemma that for every ε > 0 the series ∞ n=1 P ∞ (b n ξ n ≥ ε) converges. Consequently, lim n→∞ b n ξ n = 0 a.e. for P ∞ .
Proof. It is enough to observe that by Jensen's inequality and Lemma 2 for every x ∈ X we have , and then to apply Theorem 2 with a n = n, n ∈ N.