Continuous dependence of the weak limit of iterates of some random-valued vector functions

Given a probability space (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}$$(\Omega ,\mathcal {A},\mathbb {P})$$\end{document}, a complete separable Banach space X 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(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal B(X)$$\end{document} of all its Borel subsets, an operator Λ:Ω→L(X,X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda :\Omega \rightarrow L(X,X)$$\end{document} and ξ:Ω→X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi :\Omega \rightarrow X$$\end{document} we consider the B(X)⊗A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}(X)\otimes \mathcal A$$\end{document}-measurable function 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} given by f(x,ω)=Λ(ω)x+ξ(ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(x,\omega )=\Lambda (\omega )x+\xi (\omega )$$\end{document} and investigate the continuous dependence of a weak limit πf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi ^f$$\end{document} of the sequence of iterates (fn(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}$$(f^n(x,\cdot ))_{n\in \mathbb {N}}$$\end{document} of f, defined by f0(x,ω)=x,fn+1(x,ω)=f(fn(x,ω),ωn+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f^0(x,\omega )=x, f^{n+1}(x,\omega )=f(f^n(x,\omega ),\omega _{n+1})$$\end{document} for 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} and ω=(ω1,ω2,⋯)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega =(\omega _1,\omega _2,\dots )$$\end{document}. Moreover for X taken as a Hilbert space we characterize πf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi ^f$$\end{document} via the functional equation φf(u)=∫Ωφf(Λ(ω)u)φξ(u)P(dω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \varphi ^f(u)=\int _{\Omega }\varphi ^f(\Lambda (\omega )u)\varphi ^{\xi }(u)\mathbb {P}(d\omega ) \end{aligned}$$\end{document}with the aid of its characteristic function φf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi ^f$$\end{document}. We also indicate the continuous dependence of a solution of that equation.


Introduction
Fix a probability space (Ω, A, P) and a separable Banach space X. By B(X) we denote the family of all Borel subsets of X. A map f : X × Ω → X measurable with respect to the product algebra B(X) ⊗ A (shortly: B(X) ⊗ A-measurable) is called a random-valued function or an rv-function. By f n we denote the n-th iterate of f , given by f 0 (x, ω) = x and f n (x, ω 1 , . . . ω n ) = f (f n−1 (x, ω 1 , . . . , ω n−1 ), ω n ) (x, ω) −→ η(ω)x + ξ(ω), (1.1) where η : Ω → R, ξ : Ω → X are A-measurable. These maps are related to perpetuities, see for instance [11,12,16]); they are also applied to refinement type equations [15]. Substituting a random vector η into a random operator, we will consider rv-functions of the form where Λ(ω): X → X is a continuous and bounded operator for ω ∈ Ω. A function (1.2) will be called a generalized random affine map or GRAM, for short. However, the main motivation to study such rv-functions is the work of K. Baron [5], where a special case of map (1.2) with the same operator Λ(ω) for any ω was examined. The first aim of the present paper is to give some natural conditions under which the sequence of iterates of GRAM's f converges in law to π f , and to establish the continuity of the operator f −→ π f by showing how π f change if Λ and ξ do. This extends the main result of [3] as well as [4,Theorem 1] and [14,Theorem 5.2].
In the case when X is a real Hilbert space a characterization of a limit distribution π f by its characteristic function ϕ f via the linear functional equation ϕ f (u) = ϕ f (Λ * (u)) · ϕ ξ (u) was established in [5]. Referring to that paper we will show that the function ϕ f for GRAM's f is only one solution of the equation in a class of characteristic functions. Moreover, we will indicate continuous dependence in such a characterisation of the limit distribution. Vol. 97 (2023) Continuous dependence of the weak limit of iterates 755

Notions and basic facts
Throughout the paper (X, · ) is a separable Banach space and (Ω, A, P) is a given probability space. We write B(X) for a space of all Borel and bounded functions endowed with the supremum norm · ∞ and C(X) for its subspace containing all continuous (and bounded) functions. A space of all linear and continuous operators Λ : X → X will be denoted by L(X, X). We use the symbol M 1 (X) to denote the space of all probability measures defined on B(X). For short, we will write ϕdμ instead of X ϕ(x)μ(dx) for Bochner integrable ϕ and μ ∈ M 1 (X) if there is no confusion. We also consider a family of all measures with finite first moment given by for every B ∈ B(X).
We write μ χ to denote a probability distribution of the random variable χ.
where μ (χ,ζ) is their joint probability distribution. We say that a sequence (μ n ) of measures from M 1 (X) converges weakly to μ if fdμ n − −−− → n→∞ fdμ for every f ∈ C(X). We introduce the symbol d F M to denote the Fortet-Mourier metric (also known as the bounded Lipschitz distance) given by and additionally d H to denote the Huthinson metric given by where Note that the distance between some measures in the Huthinson metric may be infinite. It is known (see [9,Theorem 11.3.3]) that weak convergence is metrizable by the Fortet-Mourier norm.
With an rv-function f : X × Ω → X we may associate a linear operator P : M 1 (X) → M 1 (X) by the formula Komorek AEM which will be used in this paper. It can be shown that P is the Markovian transition operator for the distribution π n of f n given by By the convergence in distribution or in law of the sequence of iterates (f n (x, ·)) n∈N we mean that the sequence (π n (x, ·)) n∈N converges weakly to a probability distribution. Following [1] and [13] we consider a family of rv-functions f : X × Ω → X which satisfy: A simple criterion [13, Corollary 5.6], cf. [1, Theorem 3.1], for the convergence in distribution of iterates of rv-functions reads as follows: Then for every x ∈ X the sequence of iterates (f n (x, ·)) n∈N converges in distribution and the limit π f does not depend on x. Moreover π f ∈ M 1 1 (X) and for n ∈ N and x ∈ X.
The geometric rate of convergence allows us to formulate a result concerning the continuity of f −→ π f . We cite a part of [14,Theorem 4.1] that will be useful in the next section.

Proposition 2.2.
Assume that rv-functions f, g satisfy (H f ) and (H g ), respectively. Then for limit distributions π f and π g , occurring in Proposition 2.1, we have for h ∈ {f, g}. A similar convention will be used considering condition (U g ) in the next section.

Continuous dependence of the limit distribution of generalized random affine maps
Fix Λ : Ω → L(X, X) and A-measurable ξ : Ω → X. Since X is separable, we may consider equivalently the weak, strong (in Bochner's sense), and Borel measurability of the random variable ξ. To get some results concerning the convergence in law of GRAM's (1.2) we need to show that (1.2) is an rvfunction. To do this we will introduce the following: We call a map Λ : Ω → L(X, X) a random operator, if it is A-measurable, i.e. Λ −1 (B) ∈ A for every Borel subset B of L(X, X).
Proof. Fix x ∈ X and define ϕ x : L(X, X) → X by ϕ x (T ) = T x. It is obvious that ϕ x is linear, and since Remark 3.3. One can show that for a separable space X if Λ(·)x : Ω → X is Ameasurable for every x ∈ X and Λ(ω): X → X is continuous for every ω ∈ Ω then a map Λ : The main result of this section concerns the continuous dependence of the limit of iterates of GRAM's. We will formulate it for a family of rv-functions f : X × Ω → X which satisfy: Komorek AEM and Λ f : Ω → L(X, X) is a random operator satisfying Theorem 3.4. Assume that rv-functions f, g satisfy (U f ) and (U g ), respectively. Then the sequences of iterates (f n (x, ·)) n∈N , (g n (x, ·)) n∈N are convergent in law to the probability distributions π f , π g ∈ M 1 1 (X), respectively, the limits do not depend on x ∈ X, and α + β , Proof. At the beginning let us observe that ( By Proposition 2.1 we infer that there exist probability distributions π f , π g ∈ M 1 1 (X) such that for every x ∈ X the sequences (f n (x, ·)) n∈N , (g n (x, ·)) n∈N are convergent in law to π f , π g , respectively.
The rest of the proof runs similarly to the proof of [14, Theorem 5.2] which concerns (1.1). For the convenience of the reader we repeat the relevant computations after appropriate changes for the case of GRAM's, thus making our exposition self-contained. So fix k ∈ N and let us define Λ k : Ω ∞ → L(X, X) and Vol. 97 (2023) Continuous dependence of the weak limit of iterates 759 and • is a composition. From that Then and from the inequality we have Therefore for the function α f (x) given by (2.3) we obtain A similar inequality holds for α g (x) .
and applying Proposition 2.2 we finish the proof.

Corollary 3.5. Assume that rv-functions f, g have the form
Then the sequences of iterates (f n (x, ·)) n∈N , (g n (x, ·)) n∈N are convergent in law to the probability distributions π f , π g ∈ M 1 1 (X), respectively, the limits do not depend on x ∈ X, and

Characterisation of the limit distribution
Let (Ω, A, P) be a probability space. In this section X is a separable real Hilbert space with the inner product (·|·). However in cases when it is not needed we will emphasize it. We define a characteristic function ϕ f of the rv-function f , assuming that the iterates (f n (x, ·)) n∈N converge in law and the limit does not depend on x; in such a case we denote by π f the distribution of the limit, i.e.
Definition 4.1. A function ϕ χ : X → C given by is called a characteristic function of the X-valued random variable χ with distribution μ χ .

Definition 4.2.
A function ϕ f : X → C given by is called a characteristic function of the rv-function f . The problem of characterization of the limit distribution π f via a functional equation for its characteristic function ϕ f was considered in [5]. The author showed that for the rv-function f given by with Λ ∈ L(X, X) such that Λ < 1 and a random variable ξ : Ω → X such that E ξ < ∞ its characteristic function ϕ f is the only solution of the equation where Λ * stand for the adjoint operator to Λ, which satisfies (Λ * u|z) = (u|Λz) for every u, z ∈ X. Our goal is to generalize this result to GRAM's. First we give some preceding facts, which will be needed in the general setting.

Lemma 4.3.
Let X be a Banach space. Assume that a random operator Λ : Ω → L(X, X) and a random variable ξ : Ω → X are independent. If x ∈ X, then Λ(·)x : Ω → X and ξ : Ω → X are independent.
Proof. Fix B ∈ B(X 2 ). Put which ends the proof.
Vol. 97 (2023) Continuous dependence of the weak limit of iterates 763 Corollary 4.5. Let X be a separable Banach space. Assume that an rv-function f : X × Ω → X is given by (1.2), where Λ : Ω → L(X, X) is a random operator and ξ : Ω → X is a random variable. If Λ and ξ are independent, x ∈ X and n ∈ N, then Λ n+1 (·)f n (x, ·): Having proved independence we also have to characterise the probability distribution of the sum of independent random variables. It is well known that such a distribution can be described as the convolution of each random variable distributions. More precisely, we have: Theorem 4.6. Let X be a separable Banach space. If η : Ω → X, ξ : Ω → X are independent random variables, then for every ω ∈ Ω, x, y ∈ X is called an adjoint random operator to Λ.
Proof. According to Remark 3.3 it is enough to show that Λ * (·)x : Ω → X is A-measurable for every x ∈ X. Fix x ∈ X and observe that (x|Λ(ω)y): Ω → R is A-measurable for every y ∈ X. By the Riesz Representation Theorem for every linear functional y * : X → R there exists y such that y * Λ * (ω)x = (Λ * (ω)x|y) for every ω ∈ Ω.
Therefore from the A-measurability of (x|Λ(·)y): Ω → X we conclude that Λ * (·)x is weak measurable. Since X is separable, we may conclude that Λ * (·)x is strong measurable and consequently A-measurable.  Komorek AEM E Λ(·) < 1, E ξ < ∞. Moreover, assume that Λ and ξ are independent. Then the characteristic function ϕ f of f is the only solution of the equation which is continuous at zero, bounded and fulfills ϕ f (0) = 1.
Lemma 4.11. Let (Ω, A, P) be an arbitrary probability space. Suppose that the independent and identically distributed random variables ζ i : Ω → R, i ∈ N fulfil the following properties Then the sequence ( n i=1 ζ i ) n∈N converges a.s. to zero.
Proof. To show convergence we will consider three cases: Define a set A n = {ω ∈ Ω : n i=1 ζ i (ω) = 0} and observe that A n+1 ⊂ A n , and By the continuity of the measure it follows that III. Now assume that 0 < Eζ i < 1, and P(ζ i = 0) = 0. From Jensen's inequality If −∞ < E log ζ 1 then by the independence of ζ i s we can apply the Strong Law of Large Numbers, hence for 0 < < |E log ζ 1 | there exists N ∈ N such . . , ω n ) ∈ B}) and observe that π Λf n (x, ·) = Qπ f n (x, ·) for every x ∈ X. Indeed, for fixed x ∈ X, B ∈ B(X) it holds that Komorek AEM So now, by Corollary 4.5 and Theorem 4.6 we see that π f n+1 (x, ·) = π Λf n (x, ·) * μ ξ = Qπ f n (x, ·) * μ ξ . It can be easily shown that the Markov operator Q has the Feller property. To do this let us see at first that For a fixed ψ ∈ C(X) take an arbitrary x 0 ∈ X and note that for every Let us define ϕ n (ω) = ψ(Λ(ω)x n ) and ϕ 0 (ω) = ψ(Λ(ω)x 0 ). Since |ϕ n (ω)| ≤ ψ ∞ for ω ∈ Ω, n ∈ N we can apply the Lebesgue Dominated Convergence theorem and hence Because x 0 , (x n ) n∈N and ψ are arbitrary, we have Q * (C(X)) ⊂ C(X). From that and [18, Theorem 1.1, Ch. III] we can pass n to the limit and we obtain Now from the definition of the characteristic function we make the following computations This shows that ϕ f satisfies (4.1). It remains to show the uniqueness of the solution of (4.1). To do this, let us assume that ϕ is a bounded, continuous at zero solution of (4.1) and ϕ(0) = 1. Then observe that Vol. 97 (2023) Continuous dependence of the weak limit of iterates 767 It follows that for every n ∈ N we can write Since Λ * (ω) = Λ(ω) for every ω ∈ Ω, we have E Λ * (·) = E Λ(·) < 1.
By Lemma 4.11 we conclude that the sequence (Λ * ) n (·)(u) n∈N converges a.s. to zero. Fix n ∈ N and let us define random variables η n , θ n : Ω ∞ → C, respectively, by Hence we can rewrite (4.3) as ϕ(u) = Ω ∞ θ n (ω)η n (ω)P ∞ (dω), n ∈ N, u ∈ X and thus we obtain Observe that |θ n (ω) − 1| ≤ ϕ ∞ + 1 and (θ n ) n∈N converges a.s. to 1, by the continuity of ϕ at zero. Therefore, from the Lebesgue dominated convergence theorem it can be concluded that Komorek AEM Hence passing with n to the limit we obtain which completes the proof. To show assertion (i) observe that for a function ϕ which is a solution of (4.1) and M > 0, a Lipschitz constant of ϕ, the following inequalities hold, which yields (4.4).
When (ii) holds, the formula (4.3) reduces to for any n ∈ N. Passing with n to the limit we obtain Remark 4.13. Note that the expression (4.4) is in fact the formula of the unique solution ϕ of (4.1). In particular, when Λ is independent of ω, this solution takes the form (4.5) and it can also be found in [5,Theorem 3.1].
We now give an example of a GRAM which satisfies the assumptions of Theorem 4.10.
Example 4.14. Let us consider random variables ξ : Ω → X and κ : Ω → N. Take a countable family of linear bounded operators T i : X → X, i ∈ N. We define Λ : Ω → L(X, X) as Then the following statements hold: (i) Λ is a random operator.
Vol. 97 (2023) Continuous dependence of the weak limit of iterates 769 (ii) If ξ and κ are independent, then so are ξ and Λ.

D.
Komorek AEM Statement (iii) is obvious. Finally to show (iv) fix i ∈ N and observe that for ω ∈ κ −1 ({i}) we have (Λ * (ω)x|y) = (x|T i y) = (T * i x|y) for every x, y ∈ X. Therefore Λ * (ω) = T * i for ω ∈ κ −1 ({i}). From that we obtain By statements (i)-(iv) we can consider an rv-function f of the form and if we assume additionally that i∈N μ κ ({i}) T i < 1 and E ξ < ∞, then Theorem 4.2 allows us to claim that (provided that κ and ξ are independent) the characteristic function ϕ f is the only solution of the equation which is bounded, continuous at zero and ϕ(0) = 1.
It is worth pointing out that if we consider the class of solutions ϕ of the equation (4.1) (or in particular of (4.6)) which do not have to be either bounded or Lipschitz, then such a class can contain more than one solution, which is shown in the example given below.
Observe furthermore that Λ and Λ * have the same distribution.