The geometric rate of convergence of random iteration in the Hutchinson distance

Using the Banach fixed-point theorem we provide a simple criterion of the geometric rate of convergence and of asymptotic stability of Markov operators in the Hutchinson distance. The obtained results are applied to sequences of iterates of random-valued functions.


Introduction
In 1995 Lasota [10] established some sufficient conditions for the asymptotic stability of Markov Feller operators acting on a metric space and obtained two nice criteria for Markov operators generated by iterated function systems. The first criterion was based on the so called double contraction principle discovered by Lasota, which is a generalization of the classical Banach fixed-point theorem. Following an idea from [10] Baron has used the Hutchinson distance of distributions to get the geometric rate of convergence of sequences of distributions of iterates of random valued vector functions in the Fortet-Mourier metric (see [1]). Recently this result has been strengthened in [2] by the fact that the distribution of the limit has the first moment finite. The aim of the present paper, motivated by [1,10], is to show how fast a sequence of iterates of Markov operators tends in the Hutchinson distance to a unique invariant and attractive measure. Moreover, as an application of our main result we will examine a speed of convergence of sequences of iterates of random-valued functions having values in metric spaces and adopting a definition of iterates 150 R. Kapica AEM from [4]. To get our criterion of the geometric rate and asymptotic stability of Markov operators in the Hutchinson distance we will use the Banach fixed-point theorem. It turns out that by the use of that classical contraction mapping principle we give stronger results under the same assumptions than in the listed papers. Namely, Theorem 4.

Notions and basic facts
Assume that (X, ) is a Polish space, i.e. a separable and complete metric space. Let B(X) stand for the σ-algebra of all Borel subsets of X. By M 1 (X) we denote the space of all probability measures on B(X). Let B(X) denote the space of all bounded Borel-measurable functions equipped with the supremum norm || · || ∞ and C(X) be the subspace of bounded continuous functions. For brevity we will write ϕdμ instead of X ϕdμ, where ϕ ∈ B(X) and μ ∈ M 1 (X). Recall that a sequence of measures (μ n ) converges weakly to μ, if ϕdμ n − −−− → n→∞ ϕdμ for every ϕ ∈ C(X). It is well known (see [6,Theorem 11.3.3]) that this convergence is metrizable by the Fortet-Mourier (known also as Lévy-Prokhorov) metric [7] where  [6,11]. Throughout this paper we shall consider a regular Markov operator P : M 1 (X) → M 1 (X), i.e. P is a linear operator (here linearity is restricted to nonnegative and summing up to one coefficients only) and there exists an (adjoint or dual) operator P * : B(X) → B(X) such that ϕdP μ = P * ϕdμ for any ϕ ∈ B(X) and μ ∈ M 1 (X). Moreover, if P * : B(X) → B(X) is a linear operator, P * 1 X = 1 X and P * ϕ ≥ 0 if ϕ ≥ 0, then the operator P given by P μ(A) = P * 1 A (x)μ(dx), A ∈ B(X), is the Markov operator (with adjoint P * ).

Vol. 93 (2019)
The geometric rate of convergence of random iteration 151 Assume that (Ω, A, P) is a probability space. A function f : X × Ω → X is said to be a random-valued function (shortly rv-function) if it is measurable with respect to the product σ-algebra B(X) ⊗ A. Having an rv-function f we will examine a regular Markov operator P : (2.1) One can show that P is a transition operator for a sequence of iterates of rv-functions in the sense of Baron and Kuczma [4] (cf. [5]). More precisely, denotes the distribution of the nth iterate of f defined inductively as follows

Properties of some family of measures
Let (X, ) be a Polish space. Assume that μ 0 ∈ M 1 (X) is fixed and let us consider a family M(μ 0 ) given by Clearly, if μ ∈ M(μ 0 ), then d H (μ, ν) < ∞ if and only if ν ∈ M(μ 0 ). In other words the set M(μ 0 ) does not depend on the choice of the measure μ 0 in the following sense: (By δ x we mean the Dirac measure concentrated at x ∈ X.) We will show that M(δ x0 ) is equal to the family M 1 1 (X) defined by for some x 0 ∈ X; see Lemma 3.1 given below. Note that the set M 1 1 (X) is independent of x 0 and it consists of all Borel measures on X with the first moment finite.
To do this let us consider ϕ n ∈ Lip b 1 (X), n ∈ N, given by Obviously, the sequence (ϕ n ) converges pointwise to ϕ, and by the Lebesgue dominated theorem we have Thus we get Then V ∈ Lip 1 (X) and for every ϕ ∈ Lip 1 (X) we have Vol. 93 (2019) The geometric rate of convergence of random iteration 153 where V is given by (3.2). Hence the sequence (V n ) is increasing and convergent to V , and  Proof. The fact that d H is a (finite) metric on M(μ 0 ) was already explained, so let us turn to the proof of completeness. To do this fix a Cauchy sequence (μ n ) in (M(μ 0 ), d H ) and observe that d F M (μ n , μ m ) ≤ d H (μ n , μ m ), which shows that (μ n ) is a Cauchy sequence in a space M 1 (X) with the Fortet-Mourier metric d F M . Since this metric is complete it follows that (μ n ) converges in d F M , or equivalently (μ n ) is weakly convergent, say to μ ∈ M 1 (X). Fix ε > 0 and assume that n 0 ∈ N is such that d H (μ n , μ m ) < ε for n, m ≥ n 0 .
Then for the same n, m, and for an arbitrarily fixed ϕ ∈ Lip b 1 (X) we have Consequently for any fixed n ≥ n 0 ϕdμ − ϕdμ n ≤ ϕdμ − ϕdμ m + ε for m ≥ n 0 , and passing with m to infinity we see that ϕdμ − ϕdμ n ≤ ε for any n ≥ n 0 .

R. Kapica AEM
A proof of the density of M(μ 0 ) is, in fact, a part of a proof of [10, Theorem 3.2]. For the convenience of the reader we will show it based on ideas from that paper. Fix μ ∈ M 1 (X), x 0 ∈ X, and let r be any positive real such that μ(B r ) > 0, where B r is an open ball centered at x 0 with radius r. Define a measure μ r ∈ M 1 (X) by where c r = μ 0 (B r )/μ(B r ) and B r = X \ B r . We will show that μ r ∈ M(μ 0 ). To this end fix ϕ ∈ Lip b 1 (X) and note that Thus d H (μ r , μ 0 ) ≤ 2rμ 0 (B r ) < ∞, i.e. μ r ∈ M(μ 0 ). (Clearly μ r ∈ M(μ 0 ) for r > r.) Now for any ϕ ∈ Lip 1 (X), ||ϕ|| ∞ ≤ 1 we have 0, which finishes the proof.

Asymptotic stability and the rate of convergence of Markov operators
Let (X, ) be a Polish space. We say that a Markov operator P : M 1 (X) → M 1 (X) is asymptotically stable, if it has an invariant and attractive measure. Strictly speaking, μ * ∈ M 1 (X) is invariant for P if P μ * = μ * or alternatively P * ϕdμ * = ϕdμ * for every ϕ ∈ B(X); respectively, μ * is attractive for P if P n μ w − −−− → n→∞ μ * for every μ ∈ M 1 (X), where P 1 = P, P n+1 = P n • P Vol. 93 (2019) The geometric rate of convergence of random iteration 155 and the limit is in a weak sense. Clearly, if P is asymptotically stable then P has exactly one invariant measure. The main result of this section concerns a regular Markov operator and it reads as follows.
Theorem 4.1. Let P : M 1 (X) → M 1 (X) be a regular Markov operator with adjoint operator P * : B(X) → B(X). Assume that there exists λ ∈ (0, 1) for which for some μ 0 ∈ M 1 (X). Then the operator P is asymptotically stable and its invariant measure μ * belongs to M(μ 0 ). Moreover we have a geometric rate of convergence, i.e for any n ∈ N and μ ∈ M(μ 0 ).
The proof of Theorem 4.1 will be preceded by a few results concerning the assumptions of the theorem.
Remark 4.2. There is no need for the assumption in Theorem 4.1 as well as in the following facts of the paper that the operator P has the Feller property, i.e. P * (C(X)) ⊂ C(X), which implies that such an operator is continuous in the topology of weak convergence. The property considered here was assumed in the main results of [10].

Proposition 4.4.
Assume that an operator P with adjoint P * satisfies condition (4.1) with some λ ∈ (0, 1) and let μ 0 ∈ M 1 (X). Then the following assertions are pairwise equivalent: The above proposition is an immediate consequence of Lemma 4.3 and jointly with Remark 3.2 implies the following fact.

Corollary 4.5.
Assume that an operator P with adjoint P * satisfies condition (4.1) with some λ ∈ (0, 1). Then the following assertions are pairwise equivalent: Proof of Theorem 4.1. From Lemma 4.3 and Theorem 3.3 it follows that P restricted to M(μ 0 ) is a λ-contraction acting on a complete metric space M(μ 0 ) into itself. By the Banach fixed-point theorem there exists a unique invariant measure μ * ∈ M(μ 0 ) for P and condition (4.3) holds. It remains to prove the asymptotic stability of P . To this end observe first that P is a non-expanding operator with respect to the Fortet-Mourier metric. Indeed, if ϕ ∈ Lip 1 (X) and ||ϕ|| ∞ ≤ 1, then for any x, y ∈ X, which shows that P * ϕ ∈ Lip 1 (X), ||P * ϕ|| ∞ ≤ 1 and gives Fix μ ∈ M 1 (X) and ε > 0. According to Theorem 3.3 there is ν ∈ M(μ 0 ) such that d F M (μ, ν) < ε. By (4.3) we have d H (P n ν, μ * ) < ε for every large enough n ∈ N. Then for such n we obtain Vol. 93 (2019) The geometric rate of convergence of random iteration 157

Application to the iterates of random-valued functions
Let (X, ) be a Polish space. Assume that (Ω, A, P) is a probability space and f : X × Ω → X is an rv-function. We will examine a transition operator (2.1) for distributions π n (x, ·) of iterates (2.2). Putting we observe that π(x, ·) is a probability measure on B(X) and π(·, B) is a Borelmeasurable function for any fixed x ∈ X and B ∈ B(X). This allows us to show that the operator P * given by is adjoint to (2.1), and in addition, π n (x, B) = P * n 1 B (x) = P n δ x (B) (5.1) for any x ∈ X and B ∈ B(X).
Remark 5.1. We say that an rv-function f : X × Ω → X is P-continuous (see [3]), if for every sequence (x j ) of points from X tending to x ∈ X, the sequence (f (x j , ·)) converges in probability P to f (x, ·). One can show that P given by (2.1) with P-continuous f is a regular Markov Feller operator and it is asymptotically stable if and only if for every x ∈ X the sequence (f n (x, ·)) converges in distribution or in law (which means that (π n (x, ·)) converges weakly) and the limit μ * does not depend on x; see [9, Proposition 2.1 and Theorem 2.3]. Moreover, the measure μ * is an invariant measure for the operator P .