Abstract
We investigate the asymptotic behaviour of the sequence of forward type iterations of a given random-valued vector function on the state space being a separable and complete metric space. Assuming non-linear contraction in mean we prove that the considered sequence converges weakly to a random variable with a finite first moment and independent of the initial state. Moreover, we show that the speed of this convergence does not have to be geometric. We also present examples illustrating the result obtained.
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1 Introduction
Fix a probability space \((\Omega ,{\mathcal {A}},{\mathbb {P}})\) and a metric space X. Let \({\mathcal {B}}(X)\) denote the \(\sigma \)-algebra of all Borel subsets of X.
We say that \(f:X\times \Omega \rightarrow X\) is a random-valued function (shortly: an rv-function) if it is measurable for the product \(\sigma \)-algebra \({\mathcal {B}}(X)\otimes {\mathcal {A}}\). The iterates of such an rv-function are given by
for \(n\in {\mathbb {N}}\), \(x\in X\) and \((\omega _1,\omega _2,\ldots )\) from \(\Omega ^{\infty }\) defined as \(\Omega ^{{\mathbb {N}}}\). Note that \(f^n:X\times \Omega ^\infty \rightarrow X\) is an rv-function on the product probability space \((\Omega ^\infty ,{\mathcal {A}}^\infty , \mathbb P^\infty )\). More exactly, one can show that for \(n\in {\mathbb {N}}\) the n-th iterate \(f^n\) is measurable for \({\mathcal {B}}(X)\otimes \mathcal A_n\), where \({\mathcal {A}}_n\) denotes the \(\sigma \)-algebra of all sets of the form
with A from the product \(\sigma \)-algebra \({\mathcal {A}}^n\).
The iterates so defined were introduced independently in [4] and [9] with reference to functional equations (see e.g. [3, 7, 18]). In a broader context they form random forward iterations (see e.g. [8, 13]), also known as outer iterations (see e.g. [11]). These iterates are prototypes of random dynamical systems (see [1, Section 1.1]; cf. [22]) and they have Markov property. The family \(\{f(\cdot ,\omega ): \omega \in \Omega \}\) forms an iterated function system (IFS for abbreviation) in which functions \(f(\cdot ,\omega )\) are choosing independently with probability \({\mathbb {P}}\). A generalization of this concept, devoted to random iteration with place-dependent probabilities, can be found, e.g., in [16, Section 3] and [25]. As in the mentioned papers we will express asymptotic behaviour of our iterates by the convergence in law. In fact, this type of convergence of iterations is closely related to the asymptotic stability of Markov operators with the kernel of the form \((x,B)\mapsto \int _\Omega \mathbbm {1}_B(f(x,\omega ))\mathbb P(d\omega )\), determined by a fixed rv-function f. For details see [14] and for a more complete point of view we refer the reader to [13, 17, 20] and the references therein; we only mention here that Markovian operator P is asymptotically stable, if it has an invariant measure \(\mu ^*\), i.e., \(P\mu ^*=\mu ^*\), which attracts any probability Borel measure. There are many papers in which the convergence in law of iterates of rv-functions and the stability of Markov operators with the kernel determined by an rv-function are investigated; however usually a kind of Lipschitz contraction on the rv-function considered is assumed (see e.g. [2, 6, 8, 15]).
This paper aims to extend the results on convergence in law of iterates of random-valued functions that are mean contractive in the conventional sense to the case where only a weak (non-linear) form of the mean contractivity is assumed and examine how fast the sequence of iterates converges. The speed of convergence obtained does not have to be geometric as in the Lipschitz case. Let us mention that results [21, Theorem 9.2] and [23, Theorem 6.3.2] involve the same form of the contractivity property as that employed in the manuscript pertain to IFS with place-dependent probabilities of choosing them, but for finite many transformations and bring only asymptotic stability of IFS.
2 Preliminaries
Let \((X,\rho )\) be a complete and separable metric space. By \({\mathcal {M}}_1(X)\) we denote the set of all probability measures defined on \({\mathcal {B}}(X)\). \(\textrm{Lip}_{\alpha }(X)\) denotes the set of all real functions defined on X that meet a Lipschitz condition with a constant \(\alpha \in [0,\infty )\), and
It is well known (see [10, Theorem 11.3.3]) that the weak convergence of probability Borel measures on X is metrizable by the Fortet-Mourier metric \(d_{FM}:{\mathcal {M}}_1(X)\times \mathcal M_1(X)\rightarrow [0,\infty )\) given by
Moreover we will use the Hutchinson metric (see [12, 19]), also known as Wasserstein or Kantorovich-Rubinstein distance [24], defined by
According to [15, Lemma 3.1(i)],
where
with an arbitrarily fixed \(x_0\in X\); the definition does not depend on the choice of \(x_0\).
Remark 2.1
[see [24, Theorem 6.18], cf. [15, Theorem 3.3 and Remark 3.2]]. The metric space \(({\mathcal {M}}_1^1(X),d_H|_{\mathcal M_1^1(X)\times {\mathcal {M}}_1^1(X)})\) is complete.
Remark 2.2
[see [15, Theorem 3.3]]. The set \({\mathcal {M}}_1^1(X)\) is dense in \(({\mathcal {M}}_1(X),d_{FM})\).
3 Main result
We employ the following hypothesis.
-
(H)
\((X,\rho )\) is a separable and complete metric space and \(f:X\times \Omega \rightarrow X\) is an rv-function such that
$$\begin{aligned} \int _\Omega \rho (f(x,\omega ),f(z,\omega ))\mathbb P(d\omega )\le \psi (\rho (x,z))\quad \hbox {for }x,z\in X \end{aligned}$$with a concave function \(\psi :[0,\infty )\rightarrow [0,\infty )\).
Remark 3.1
If \(\psi :[0,\infty )\rightarrow [0,\infty )\) is concave, then it is non-decreasing.
Proof
Suppose, towards a contradiction, that there are \(t_1,t_2\in [0,\infty )\) such that \(t_1<t_2\) and \(\psi (t_2)<\psi (t_1)\). Fix \(\alpha \in (\frac{\psi (t_2)}{\psi (t_1)},1)\) and put \(t=\frac{t_2-\alpha t_1}{1-\alpha }\). Then
and hence \(\psi (t)\le \frac{\psi (t_2)-\alpha \psi (t_1)}{1-\alpha }<0\), a contradiction. \(\square \)
Proposition 3.1
Assume (H) and define \(P:\mathcal M_1(X)\rightarrow {\mathcal {M}}_1(X)\) by
Then
Proof
Observe first that for any Borel \(\varphi :X\rightarrow {\mathbb {R}}\), which is non-negative or bounded, we have
Fix \(\mu ,\nu \in {\mathcal {M}}_1^1(X)\) and denote by \(\Lambda (\mu ,\nu )\) the collection of all probability Borel measures \(\lambda \) on \(X\times X\) such that
If \(\lambda \in \Lambda (\mu ,\nu )\), then
Therefore, the formula
defines a functional \(T:\Lambda (\mu ,\nu )\rightarrow [0,\infty )\).
If \(\lambda \in \Lambda (\mu ,\nu )\) and \(\varphi \in \mathrm Lip_1^b({ X})\), then by (3), (H) and Jensen’s inequality (see [10, 10.2.6]), we have
and hence
Applying (1) and the Kantorovich-Rubinstein Theorem (see [10, Theorem 11.8.2]) we conclude that there exists \(\lambda _0\in \Lambda (\mu ,\nu )\) such that
This jointly with (4) implies
and the proof is complete. \(\square \)
Corollary 3.1
Assume (H) and let \(P:\mathcal M_1(X)\rightarrow {\mathcal {M}}_1(X)\) be the operator given by (2). If there exists \(x_0\in X\) such that
then \(P({\mathcal {M}}_1^1(X))\subset {\mathcal {M}}_1^1(X)\) and for every \(n\in {\mathbb {N}}\) we have
Proof
If \(\mu \in {\mathcal {M}}_1^1(X)\), then by (3) with \(\varphi =\rho (\cdot ,x_0)\) we obtain
Thus (5) implies \(P\mu \in {\mathcal {M}}_1^1(X)\).
By Proposition 3.1 we see that (6) holds for \(n=1\). If (6) holds for some \(n\in {\mathbb {N}}\), then Proposition 3.1 and Remark 3.1 for \(\mu ,\nu \in {\mathcal {M}}_1^1(X)\) imply
which completes the proof. \(\square \)
Given an rv-function \(f:X\times \Omega \rightarrow X\) we denote by \(\pi ^f_n(x,\cdot )\) the distribution of \(f^n(x,\cdot )\), i.e.,
Clearly, for every \(x\in X\), \(\pi ^f_0(x,\cdot )=\delta _x\), the Dirac measure concentrated at x, and \(\pi ^f_1(x,\cdot )\) is the distribution of \(f(x,\cdot )\). One can check that \(\pi ^f_{n+1}(x,\cdot )= P\pi _f^n(x,\cdot )\) holds for \(x\in X\) and \(\mu \in {\mathcal {M}}_1(X)\) (see [14]), which implies \(\pi ^f_{n+1}(\mu ,\cdot ) = P\pi ^f_n(\mu ,\cdot )\) for any \(\mu \in {\mathcal {M}}_1(X)\), with \(\pi ^f_n(\mu ,\cdot )=\int _X \pi ^f_n(x,\cdot )\mu (dx)\), and shows that operator P given by (2) is the transition operator of the sequence of iterates under consideration. It turns out that this operator is asymptotically stable. In fact, the following theorem gives what follows.
Theorem 3.1
Assume (H) with \(\psi \) satisfying also
If (5) holds with some \(x_0\in X\), then the operator \(P:{\mathcal {M}}_1(X)\rightarrow {\mathcal {M}}_1(X)\) given by (2) admits an invariant measure \(\pi ^f\in {\mathcal {M}}_1^1(X)\) and
Moreover,
Proof
From Corollary 3.1, Remark 2.1 and the Boyd-Wong Theorem (see [5, Theorem 1]) we conclude that there exists a measure \(\pi ^f\in {\mathcal {M}}_1^1(X)\) such that
and
To get the first inequality in (9), observe that by a simple induction for all \(x\in X\) and \(n\in {\mathbb {N}}\cup \{0\}\) we have
indeed, \(P^0\delta _x=\delta _x=\pi _0^f(x,\cdot )\) and if (11) holds for some \(n\in {\mathbb {N}}\cup \{0\}\), then
for every \(B\in {\mathcal {B}}\)(X) (cf. [14, Proposition 2.1]). Since \(\delta _x\in {\mathcal {M}}_1^1(X)\) for every \(x\in X\), Corollary 3.1 and (11) imply \(\pi ^f_n(x,\cdot )\in \mathcal M_1^1(X)\) for all \(x\in X\) and \(n\in {\mathbb {N}}\). Therefore, the first inequality in (9) follows from (8) and (11).
For the prove of the second inequality in (9), note that if \(x\in X\) and \(\varphi \in \mathrm Lip_1^b({ X})\), then
Hence
which jointly with Remark 3.1 gives the second inequality in (9).
It remains to prove the moreover part. For this purpose, we fix \(\mu \in {\mathcal {M}}_1(X)\).
If \(\varphi \in \mathrm Lip_1({ X})\) and \(|\varphi (x)|\le 1\) for every \(x\in X\), then due to (H) and (7) the function \(\phi :X\rightarrow {\mathbb {R}}\) given by
belongs to \(\mathrm Lip_1({ X})\) and \(|\phi (x)|\le 1\) for every \(x\in X\), and moreover, by (3), for every \(\nu \in {\mathcal {M}}_1(X)\) we have
whence
Fix \(\varepsilon >0\). By Remark 2.2 there exists \(\nu \in \mathcal M_1^1(X)\) such that \(d_{FM}(\mu ,\nu )\le \frac{\varepsilon }{2}\) and by (10) there exists \(n_0\in {\mathbb {N}}\) such that \(d_H(P^n\nu ,\pi ^f)\le \frac{\varepsilon }{2}\) for every \(n\ge n_0\). Hence
for every \(n\ge n_0\). \(\square \)
Remark 3.2
Assume \(f:X\times \Omega \rightarrow X\) is an rv-function. If for some \(x \in X\) the sequence \((\pi _n^f(x,\cdot ))_{n\in {\mathbb {N}}}\) converges weakly to a \(\pi \), then \(\textrm{supp}\pi \subset \textrm{cl}f(X \times \Omega )\).
Proof
Suppose there exists \(x_0 \in \textrm{supp}\pi \cap (X{\setminus }\textrm{cl}f(X \times \Omega ))\) and let B be a closed ball in X with center at \(x_0\) and contained in \(X{\setminus }\textrm{cl}f(X\times \Omega )\). Since \(x_0\in \textrm{supp}\pi \), it follows that \(\pi (B)>0\), and by Urysohn’s lemma there is a continuous \(\varphi :X \rightarrow [0,1]\) such that
Then \(\varphi \big (f^n(x,\omega )\big )=0\) for \(n \in {\mathbb {N}}\) and \(\omega \in \Omega ^\infty \), and
Hence
a contradiction. \(\square \)
By Remark 3.1 we can set
and consider \(\psi \) as a mapping of \([0,\infty ]\) into itself.
Remark 3.3
Since by the Remark 3.2 the support of the invariant measure \(\pi ^f\) from Theorem 3.1 is included in the closure of \(f(X\times \Omega )\), it follows that for all \(x\in X\) and \(y\in f(X\times \Omega )\), we have
Therefore, (9) yields
for all \(x\in X\) and \(n\in {\mathbb {N}}\).
4 Examples
Fix \(\xi \in L^1(\Omega ,{\mathcal {A}},{\mathbb {P}})\), a non-zero \(\eta \in L^1(\Omega ,{\mathcal {A}},{\mathbb {P}})\), and put \(\alpha =\frac{1}{\Vert \eta \Vert _1}\). Let \(\psi :[0,\infty )\rightarrow [0,\infty )\) be a concave function such that \(\eta \psi (x)+\xi \) is non-negative for every \(x\in [0,\infty )\) and
Note that \(\psi \) is non-expansive and \(\psi (0)=0\). In particular, the formula
defines a random affine map \(f:[0,\infty )\times \Omega \rightarrow [0,\infty )\). It is clear that (5) holds with any \(x_0\in [0,\infty )\) and
In consequence (H) holds, and one can apply Theorem 3.1. Note that neither [2, Theorem 3.1] nor [8, Theorem 1.1] do not apply, whenever
Let \(\pi ^f\) be the measure resulting from Theorem 3.1. Then (9) yields
Example 4.1
Consider \(\psi :[0,\infty )\rightarrow [0,\infty )\) given by
It is easy to see that \(\psi \) is concave, satisfies (12) with \(\alpha =1\), (13) holds, and (14) leads to
Example 4.2
Consider now \(\psi :[0,\infty )\rightarrow [0,\infty )\) given by
It is easy to check that \(\psi \) is concave, satisfies (12) with \(\alpha =1\), and (13) holds. Now, (14) gives
Applying [18, Theorem 1.3.6] we conclude that for every \(c\in \left( \sqrt{\frac{3}{2}},\infty \right) \) there exists \(n_0\in {\mathbb {N}}\) such that
Example 4.3
Fix \(\psi _0:[0,\infty )\rightarrow [0,\infty )\) of the form \(\psi _0(t)=\frac{t}{1+t}\) or \(\psi _0(t)=\arctan t\), and consider \(\psi :[0,\infty )\rightarrow [0,\infty )\) given by
Note that \(\psi \) is concave and (13) holds. Observe also that \(\psi \) satisfies (12) with \(\alpha =2\); indeed, for all \(x\ne z\) we have
Since \(\psi \le \psi _0\), we conclude that (14) implies
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Acknowledgements
The first and third authors were supported by the Institute of Mathematics of the University of Silesia (Iterative Functional Equations and Real Analysis program). The second author was supported by the Faculty of Applied Mathematics AGH UST statutory tasks within subsidy of Ministry of Education and Science.
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Baron, K., Kapica, R. & Morawiec, J. Convergence in Law of Iterates of Weakly Contractive in Mean Random-Valued Functions. Results Math 79, 69 (2024). https://doi.org/10.1007/s00025-023-02093-0
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DOI: https://doi.org/10.1007/s00025-023-02093-0