Abstract
Assume \( (\Omega , {\mathscr {A}}, P) \) is a probability space, X is a compact metric space with the \( \sigma \)-algebra \( {\mathscr {B}} \) of all its Borel subsets and \( f: X \times \Omega \rightarrow X \) is \( {\mathscr {B}} \otimes {\mathscr {A}} \)-measurable and contractive in mean. We consider the sequence of iterates of f defined on \( X \times \Omega ^{{\mathbb {N}}}\) by \(f^0(x, \omega ) = x\) and \( f^n(x, \omega ) = f\big (f^{n-1}(x, \omega ), \omega _n\big )\) for \(n \in {\mathbb {N}}\), and its weak limit \(\pi \). We show that if \(\psi :X \rightarrow {\mathbb {R}}\) is continuous, then for every \( x \in X \) the sequence \(\left( \frac{1}{n}\sum _{k=1}^n \psi \big (f^k(x,\cdot )\big )\right) _{n \in {\mathbb {N}}}\) converges almost surely to \(\int _X\psi d\pi \). In fact, we are focusing on the case where the metric space is complete and separable.
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1 Introduction
Fix a probability space \((\Omega ,{{\mathscr {A}}},P)\) and a metric space X.
Let \( {\mathscr {B}} \) 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 with respect to the product \( \sigma \)-algebra \( {\mathscr {B}} \otimes {\mathscr {A}} \). 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 \( (\Omega ^{\infty }, {\mathscr {A}}^{\infty }, P^{\infty } ) \). More exactly, for \(n \in {\mathbb {N}}\) the n-th iterate \( f^n \) is \( {\mathscr {B}} \otimes {\mathscr {A}}_n \)-measurable, where \( {\mathscr {A}}_n \) denotes the \( \sigma \)-algebra of all sets of the form
with A from the product \( \sigma \)-algebra \( {\mathscr {A}}^n \). See [10, Sec. 1.4], [8].
A result on a.s. convergence of \( \big (f^n(x, \cdot )\big )_{n \in {\mathbb {N}}} \) 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 \(\big (f^n (x, \cdot )\big )_{n \in {\mathbb {N}}} \) to a random variable independent of \( x \in X \) 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 \(({\mathscr {F}} _n)_{n \in {\mathbb {N}}}\) be an increasing sequence of sub-\(\sigma \)-algebras of \({\mathscr {A}} \) and \((\xi _n)_{n\in {\mathbb {N}}}\) a sequence of random variables such that \(\xi _n\) is \({\mathscr {F}} _n\)-measurable and \({\mathbb {E}}(\xi _{n+1}|{\mathscr {F}} _n)=0 \) for each \(\ n \in {\mathbb {N}}\). If \((a_n)_{n \in {\mathbb {N}}}\) is an increasing and unbounded sequence of positive reals and
then
2 A Scheme
Assume X is a metric space and \( f: X \times \Omega \rightarrow X \) an rv-function.
Lemma 1
If \(\varphi :X \rightarrow {\mathbb {R}}\) is Borel and \(\varphi \circ f^n(x,\cdot )\) is integrable for \(P^\infty \) for each \(x \in X\) and \(n \in {\mathbb {N}}\), then the function \(\alpha :X \rightarrow {\mathbb {R}}\) defined by
is Borel and
Proof
Since \(\varphi \circ f\) is \( {\mathscr {B}} \otimes {\mathscr {A}} \)-measurable, by Fubini’s theorem \(\alpha \) is Borel. Consequently, for every \(x \in X\) and \(n \in {\mathbb {N}}\) the function \(\alpha \circ f^n(x,\cdot )\) is \({\mathscr {A}}_n \)-measurable and for each \( A \in {\mathscr {A}}^n\) we have
\(\square \)
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 \( \psi :X \rightarrow {\mathbb {R}}\) and assume that there exists a Borel and bounded \(\varphi :X \rightarrow {\mathbb {R}}\) such that
If \((a_n)_{n \in {\mathbb {N}}}\) is an increasing and unbounded sequence of positive reals such that
then, for every \(x \in X\),
Proof
Define \(\alpha :X \rightarrow {\mathbb {R}}\) by (1). Since \(\varphi \) is bounded, \(|\varphi (x)|\le M \) for every \(x \in X\) with an \(M \in (0,\infty )\). Obviously also \(|\alpha (x)|\le M \) for every \(x \in X\). Fix \(x \in X\) and put
Then \(|\xi _n|\le 2M\) and by Lemma 1, \({\mathbb {E}}(\xi _{n+1}|{\mathscr {A}} _n)=0 \) for each \(n \in {\mathbb {N}}\). It now follows from Brunk-Prokhorov-type theorem (C) that
Since \(\psi =\varphi -\alpha \), for every \( n \in {\mathbb {N}}\) we have
i.e.,
for every \( n \in {\mathbb {N}}\). Moreover, \(|\alpha \circ f^n(x,\cdot )|\le M\). Consequently (3) holds. \(\square \)
3 The Weak Limit
Assume now the following hypothesis (H).
(H) \((X,\rho )\) is a complete and separable metric space and \( f: X \times \Omega \rightarrow X \) is an rv-function such that
with a \( \lambda \in (0,1) \), and
Then (see [1, Theorem 3.1]) there exists a probability Borel measure \( \pi ^f\) on X such that for every \(x \in X\) the sequence of distributions of \(f^n(x,\cdot ), \ n \in {\mathbb {N}}\), converges weakly to \(\pi ^f\). See also [3, Lemma 2.2] and [9, Corollary 5.6 and Lemma 3.1].
This limit distribution \( \pi ^f\) plays an important role in solving functional equations, in particular in the class of Hölder continuous functions. We call a function \(\psi :X \rightarrow {\mathbb {R}} \) Hölder continuous with exponent \(\delta \in (0,1]\) if there is a constant \(L \in [0,\infty )\) such that
Moreover we call a function Hölder continuous if it is Hölder continuous with an exponent \(\delta \in (0,1]\). The following theorem (see [3, Theorem 2.1] and [4, Corollary 2.6]) will be useful to us.
(B) Assume (H). If \(\psi :X \rightarrow {\mathbb {R}}\) is Hölder continuous with exponent \(\delta \in (0,1]\), then it is integrable for \( \pi ^f\) and if additionally
then there exists a Hölder continuous with exponent \(\delta \) function \(\varphi :X \rightarrow {\mathbb {R}}\) such that (2) holds.
4 Main Results
In what follows \((X,\rho )\) is a metric space and \( f: X \times \Omega \rightarrow X \) 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.
Theorem 1
If \((X,\rho )\) is complete and separable with finite diameter and (7) holds with a \(\lambda \in (0,1)\), then for every Hölder continuous \(\psi :X \rightarrow {\mathbb {R}}\) and for each \(x \in X\),
Proof
Fix a Hölder continuous \(\psi :X \rightarrow {\mathbb {R}}\). Replacing \(\psi \) by \(\psi -\int _X\psi d\pi ^f\) we may assume that (9) holds. By (B) there is a Hölder continuous \(\varphi :X \rightarrow {\mathbb {R}}\) satisfying (2). Since X is bounded, so is \(\varphi \). Applying now Proposition 1 with \(a_n=n\) for \(n \in {\mathbb {N}}\) we obtain (3) which ends the proof. \(\square \)
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.
Corollary 1
If \((X,\rho )\) is compact and (7) holds with a \( \lambda \in (0,1)\), then we have (10) for every continuous \(\psi :X \rightarrow {\mathbb {R}} \) and for each \(x \in X\).
Theorem 2
Assume (H). Let \(x \in X\) and
with a \(\delta \in (0,1]\) and an increasing and unbounded sequence \((a_n)_{n \in {\mathbb {N}}}\) of positive reals. If \(\psi :X \rightarrow {\mathbb {R}}\) is Hölder continuous with exponent \(\delta \), then
The proof will be based on three lemmas.
Assume that \((X,\rho )\) is separable, (7) holds with a \(\lambda \in (0,1)\), (8) is satisfied and \(\varphi :X \rightarrow {\mathbb {R}}\) is Hölder continuous with exponent \(\delta \in (0,1]\), i.e.,
with an \(L \in [0,\infty )\).
Lemma 2
For every \(x \in X\) and \(n \in {\mathbb {N}}\) we have
Proof
Fix \(x \in X, \ n \in {\mathbb {N}}\) 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 \(x \in X\) and \(n \in {\mathbb {N}}\) we have
\(\square \)
Lemma 2 makes sense to define a Borel function \(\alpha :X \rightarrow {\mathbb {R}}\) by (1).
Lemma 3
For every \(x \in X\) and \(n \in {\mathbb {N}}\) we have
Proof
Since, for every \(\omega \in \Omega ^\infty \) and \(\omega ' \in \Omega \),
for every \(\omega \in \Omega \) we have
Hence, applying Jensen’s inequality and Fubini’s theorem,
\(\square \)
Lemma 4
Let \((b_n)_{n \in {\mathbb {N}}}\) be a converging to zero sequence of positive reals. If \(x \in X\) and there is a \(p \in (0,\infty )\) such that
then
Proof
If \(n \in {\mathbb {N}}\) and \(\omega \in \Omega \), then by (1), (12), Jensen’s inequality and (7) we have
Now to finish the proof it is enough to show that \(\lim _{n \rightarrow \infty }b_n\xi _ n=0\) a.e. for \(P^\infty \), where \(\xi _n=\rho \big (f^n(x,\cdot ),x\big )^\delta \) for \(n \in {\mathbb {N}}\). To this end observe that by Markov’s inequality for every \(n \in {\mathbb {N}}\) and \(\varepsilon > 0\) we have
Hence it follows from the assumption of the lemma that for every \(\varepsilon > 0\) the series \(\sum _{n=1}^\infty P^\infty (b_n\xi _n\ge \varepsilon )\) converges. Consequently, \(\lim _{n \rightarrow \infty }b_n\xi _ n=0\) a.e. for \(P^\infty \). \(\square \)
Proof of Theorem 2. Fix a Hölder continuous with exponent \(\delta \) function \(\psi :X \rightarrow {\mathbb {R}}\). Replacing \(\psi \) by \(\psi -\int _X\psi d\pi ^f\) we may assume that (9) holds. By (B) there is a Hölder continuous with exponent \(\delta \) function \(\varphi :X \rightarrow {\mathbb {R}}\) satisfying (2). Now using Lemma 2 define a Borel function \(\alpha :X \rightarrow {\mathbb {R}}\) by (1). Since \(\psi =\varphi -\alpha \), (6) follows. Applying Lemmas 1 and 3, and the Brunk-Prokhorov-type theorem (C) to the sequence of random variables \((\xi _n)_{n \in {\mathbb {N}}}\) defined by (4), we have (5). Finally, by Lemma 4 with \(b_n=\frac{1}{a_n}, \ n \in {\mathbb {N}}\), and \(p=2\),
This, (5), (6) and (9) give (11). \(\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \Box \)
Corollary 2
Assume (H). If \(\psi :X \rightarrow {\mathbb {R}}\) is Hölder continuous with an exponent \(\delta \le \frac{1}{2}\), then we have (10) for each \(x \in X\).
Proof
It is enough to observe that by Jensen’s inequality and Lemma 2 for every \(x \in X\) we have
and then to apply Theorem 2 with \(a_n=n, \ n \in {\mathbb {N}}\). \(\square \)
To get a result for exponents \(\delta > \frac{1}{2}\) we accept the following hypothesis (H\(_\delta \)) with parameter \(\delta \in (0,\infty )\).
(H\(_\delta \)) \((X,\rho )\) is a complete and separable metric space, \(f: X \times \Omega \rightarrow X \) is an rv-function such that
where \(\xi :\Omega \rightarrow [0,\infty )\) is a random variable for which \({\mathbb {E}}(\xi ^{2\delta })<1\), and
with an \(x_0 \in X\).
Remark 1
If \(\delta \ge \frac{1}{2}\), then (H\(_\delta \)) implies (H).
Proof
Assume (H\(_\delta \)) with a \(\delta \ge \frac{1}{2}\). By Jensen’s inequality
and
Moreover, for every \(x \in X\),
\(\square \)
Theorem 3
Assume (H\(_\delta \)) with a \(\delta \in [\frac{1}{2},1]\). If \(\psi :X \rightarrow {\mathbb {R}}\) is Hölder continuous with exponent \(\delta \), then we have (10) for each \(x \in X\).
Proof
By Remark 1 we have (H), and it follows from Theorem 2 that to finish the proof it is enough to show that for every \(x \in X\) the sequence
is bounded. This follows from the lemma that is stated below. \(\square \)
Let
Lemma 5
Assume (13) holds with a random variable \(\xi :\Omega \rightarrow [0,\infty ) \) and let p be a positive real. If \({\mathbb {E}}(\xi ^p)<1\) and \(\beta _p (x_0) < \infty \) for an \(x_0 \in X\), then \(\beta _p (x) < \infty \) for every \(x \in X\) and there exists a constant \(c_p \in (0,\infty )\) such that
Proof
Fix \(x \in X\). By (13) for every \(\omega \in \Omega \) we have
whence
Put now
and
for \(n \in {\mathbb {N}}\) and \((\omega _1, \omega _2, \ldots ) \in \Omega ^\infty \). Then, by induction and (13),
where \(\prod _{j=n+1}^n\xi _j(\omega ):=1\). Consequently,
Moreover, for every integer \(n\ge 2\) and \(k \in \{1,\ldots ,n-1\}\) the random variables \(\eta _k\), \(\xi _{k+1},\dots ,\xi _n\) are independent. Hence, if \(p \in (0,1)\), then for every \(n \in {\mathbb {N}}\) we have
If \(p \in [1,\infty )\), then by Minkowski’s inequality for every \(n \in {\mathbb {N}}\) we have
\(\square \)
Corollary 3
Assume that either
-
(i)
(H\(_\delta \)) holds with a \(\delta \in [\frac{1}{2},1]\) and \(\psi :X \rightarrow {\mathbb {R}}\) is Hölder continuous with exponent \(\delta \),
or
-
(ii)
(H\(_\frac{1}{2}\)) is satisfied and \(\psi :X \rightarrow {\mathbb {R}}\) is Hölder continuous with an exponent \(\delta \le \frac{1}{2} \).
Then for every bounded and nonempty \(A \subset X\) and for almost all \(\omega \in \Omega ^\infty \) with respect to \(P^\infty \),
$$\begin{aligned} \lim \limits _{n \rightarrow \infty }\sup \big \{\big |\frac{1}{n}\sum _{k=1}^n\psi \big (f^k(x,\omega )\big )-\int _X\psi d\pi ^f\big |: x \in A\big \}=0. \end{aligned}$$
Proof
It concerns both, (i) and (ii).
By induction,
for \(x,z \in X, \ \omega \in \Omega ^\infty \) and \(n \in {\mathbb {N}}\), with
Hence
for \(x,z \in X, \ \omega \in \Omega ^\infty \) and \(n \in {\mathbb {N}}\), with an \(L \in (0,\infty )\).
Fix \(z \in X\). Since, for every \(x \in X, \ \omega \in \Omega ^\infty \) and \(n \in {\mathbb {N}}\),
for every \(r \in (0,\infty )\) and for every nonempty subset A of the ball with center at z and radius r, for every \(\omega \in \Omega ^\infty \) and \(n \in {\mathbb {N}}\) we have
In view of Theorem 3 and Corollary 2, to finish the proof it is enough to show that
To this end observe that, by Jensen’s inequality, in the first case (i) we have
and in the second one
Therefore, applying the monotone convergence theorem and independence of \(\xi _n, \ n \in {\mathbb {N}}\), we get
Consequently, the series \(\sum _{n=1}^\infty \prod _{k=1}^n \xi _k^\delta \) converges a.e. for \(P^\infty \) and (14) follows.
\(\square \)
5 An Application to Random Affine Maps
Corollary 4
Assume X is a closed subset of a separable Banach space containing the origin, \(\xi :\Omega \rightarrow {\mathbb {R}}\) and \(\eta :\Omega \rightarrow X\) are random variables such that \(\xi (\omega )X+\eta (\omega ) \subset X\) for \(\omega \in \Omega \), and
If either \(\delta \in (0,\frac{1}{2}]\) and
or \(\delta \in [\frac{1}{2},1]\) and
then there exists a probability Borel measure \(\mu \) on X such that
and for every Hölder continuous with exponent \(\delta \) function \(\psi :X \rightarrow {\mathbb {R}}\),
Proof
The function \( f: X \times \Omega \rightarrow X \) defined by
is an rv-function. It satisfies (H) in the first case, and (H\(_\delta \)) in the second one. By induction,
for \(x \in X, \ (\omega _1,\omega _2,\ldots ) \in \Omega ^\infty \) and \(n \in {\mathbb {N}}\). Hence, \(\zeta _n=f^n(0,\cdot )\) for \(n \in {\mathbb {N}}\), so an application of Corollary 2 and Theorem 3 finishes the proof. \(\square \)
Remark 2
Let \(\lambda \in (0,1)\) and let \(\eta :\Omega \rightarrow [0,1-\lambda ]\) be a random variable. Put
for \((\omega _1,\omega _2,\ldots ) \in \Omega ^\infty \) and \(n \in {\mathbb {N}}\). By Corollary 4 there exists a probability Borel measure \(\mu \) on [0, 1] such that for every Hölder continuous \(\psi :[0,1] \rightarrow {\mathbb {R}}\),
But, as observed in [2, Remark 4.3], if \((\psi \circ \zeta _n)_{n \in {\mathbb {N}}}\) converges in probability for a Borel \(\psi : [0,1] \rightarrow {\mathbb {R}}\) such that
with a constant \(c \in (0,\infty )\), then \(\eta \) is a.s. for P constant.
References
Baron, K.: On the convergence in law of iterates of random-valued functions. Aust. J. Math. Anal. Appl. 6(1), 9 (2009)
Baron, K.: Weak law of large numbers for iterates of random-valued functions. Aequ. Math. 93, 415–423 (2019)
Baron, K.: Weak limit of iterates of some random-valued functions and its application. Aequ. Math. 94, 415–425; 427 (Correction) (2020)
Baron, K.: Continuous solutions to two iterative functional equations. Aequ. Math. 95, 1157–1168 (2021)
Dudley, R.M.: Real Analysis and Probability. Cambridge Studies in Advanced Mathematics, vol. 74. Cambridge University Press, Cambridge (2002)
Fazekas, I., Klesov, O.: A general approach to the strong law of large numbers. Theory Probab. Appl. 45, 436–449 (2000) and Teor. Veroyatn. Primen. 45, 568–583 (2000)
Kapica, R.: Convergence of sequences of iterates of random-valued vector functions. Colloq. Math. 97, 1–6 (2003)
Kapica, R.: Sequences of iterates of random-valued vector functions and solutions of related equations. Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 213, 113–118 (2004)
Kapica, R.: The geometric rate of convergence of random iteration in the Hutchinson distance. Aequ. Math. 93, 149–160 (2019)
Kuczma, M., Choczewski, B., Ger, R.: Iterative Functional Equations. Encyclopedia of Mathematics and its Applications, vol. 32. Cambridge University Press, Cambridge (1990)
Stout, W.F.: Almost Sure Convergence. Probability and Mathematical Statistics, vol. 24. Academic Press, New York (1974)
Acknowledgements
The research of Karol Baron was supported by the Institute of Mathematics of the University of Silesia in Katowice (Iterative Functional Equations and Real Analysis program). The research of Rafał Kapica was supported by the Faculty of Applied Mathematics AGH UST statutory tasks within subsidy of Ministry of Science and Higher Education.
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Baron, K., Kapica, R. Strong Law of Large Numbers for Iterates of Some Random-Valued Functions. Results Math 77, 50 (2022). https://doi.org/10.1007/s00025-021-01586-0
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DOI: https://doi.org/10.1007/s00025-021-01586-0