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

Given a probability space $$ (\Omega , {\mathcal {A}}, P) $$(Ω,A,P), a complete and separable metric space X with the $$ \sigma $$σ-algebra $$ {\mathcal {B}} $$B of all its Borel subsets and a $$ {\mathcal {B}} \otimes {\mathcal {A}} $$B⊗A-measurable $$ f: X \times \Omega \rightarrow X $$f:X×Ω→X we consider its iterates $$ f^n$$fn defined on $$ X \times \Omega ^{{\mathbb {N}}}$$X×ΩN by $$f^0(x, \omega ) = x$$f0(x,ω)=x and $$ f^n(x, \omega ) = f\big (f^{n-1}(x, \omega ), \omega _n\big )$$fn(x,ω)=f(fn-1(x,ω),ωn) for $$n \in {\mathbb {N}}$$n∈N and provide a simple criterion for the existence of a probability Borel measure $$\pi $$π on X such that for every $$ x \in X $$x∈X and for every Lipschitz and bounded $$\psi :X \rightarrow {\mathbb {R}}$$ψ:X→R the sequence $$\left( \frac{1}{n}\sum _{k=0}^{n-1} \psi \big (f^k(x,\cdot )\big )\right) _{n \in {\mathbb {N}}}$$1n∑k=0n-1ψ(fk(x,·))n∈N converges in probability to $$\int _X\psi (y)\pi (dy)$$∫Xψ(y)π(dy).


Introduction
Fix a probability space (Ω, A, P ) and a complete and separable metric space (X, ρ).
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 f 0 (x, ω 1 , ω 2 , . . .) = x, f n (x, ω 1 , ω 2 , . . .) = f f n−1 (x, ω 1 , ω 2 , . . .), ω n for n ∈ N, x ∈ X and (ω 1 , ω 2 , . . .) from Ω ∞ defined as Ω N . Note that f n : X × Ω ∞ → X is an rv-function on the product probability space (Ω ∞ , A ∞ , P ∞ ). More exactly, for n ∈ N the nth iterate f n is B ⊗ A n -measurable, where A n K. Baron AEM denotes the σ-algebra of all sets of the form with A from the product σ-algebra A n . (See [4], [5,Sec. 1.4].) A result on the a.s. convergence of f n (x, ·) n∈N for X being the unit interval may be found in [5, Sec. 1.4B]. The paper [4] by Rafa l Kapica 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 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] and applied to the equation with ϕ as the unknown function. In [2] this criterion was applied to the equation In the present paper it is strengthened and applied to get a weak law of large numbers for iterates of random-valued functions.

Wasserstein metric
By a distribution (on X) we mean any probability measure defined on B. Recall that a sequence (π n ) n∈N of distributions converges weakly to a distribution π if lim n→∞ X u(x)π n (dx) = X u(x)π(dx) for any continuous and bounded u : X → R. It is well known (see [3,Th. 11.3.3]) that this convergence is metrizable by the (Fortet-Mourier, Lévy-Prohorov, Wasserstein) metric

Convergence in law
Fix an rv-function f : X × Ω → X and let π n (x, ·) denote the distribution of f n (x, ·), i.e., with a λ ∈ (0, 1), and then there exists a distribution π on X such that for every x ∈ X the sequence π n (x, ·) n∈N converges weakly to π; moreover, Proof. It follows from [1, Th. 3.1] that there exists a distribution π on X such that (3) holds. We shall show that (4) is also satisfied. To this end note first that by (1) we have Since, by (3), for n ∈ N and by the monotone convergence theorem X ρ(x, y)π(dy) = lim n→∞ X τ n ρ(x, y) π(dy), it is enough to prove that the sequence X τ n ρ(x, y) π n (x, dy) n∈N , i.e., the To show it observe that for every n ∈ N and (ω 1 , ω 2 , . . .) ∈ Ω ∞ we have K. Baron AEM and for every y ∈ X the value f n (y, ω 1 , ω 2 , . . .) depends only on y and on (ω 1 , . . . , ω n ). Hence, applying the Fubini theorem and (5), for every n ∈ N we get Remark 3.2. If (1) holds with a λ ∈ (0, ∞) and (2) is satisfied, then the function υ : is Lipschitz.

Weak law of large numbers
Theorem 4.1. If (1) holds with a λ ∈ (0, 1) and (2) is satisfied, then there exists a distribution 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 • f k (x, ·) n∈N converges in probability to X ψ(y)π(dy).
Proof. Making use of Theorem 3.1 let π be a distribution on X such that (3) and (4) hold. It follows from Remark 3.2 and (4) that X υ(y)π(dy) < ∞.