Abstract
Given a probability space \( (\Omega , {\mathcal {A}}, P) \), a complete and separable metric space X with the \( \sigma \)-algebra \( {\mathcal {B}} \) of all its Borel subsets, a \( {\mathcal {B}} \otimes {\mathcal {A}} \)-measurable and contractive in mean \( f: X \times \Omega \rightarrow X \), and a Lipschitz F mapping X into a separable Banach space Y we characterize the solvability of the equation
in the class of Lipschitz functions \(\varphi : X \rightarrow Y\) with the aid of the weak limit \(\pi ^f\) of the sequence of iterates \(\left( f^n(x,\cdot )\right) _{n \in {\mathbb {N}}}\) of f, defined on \( X \times \Omega ^{{\mathbb {N}}}\) by \(f^0(x, \omega ) = x\) and \( f^n(x, \omega ) = f\left( f^{n-1}(x, \omega ), \omega _n\right) \) for \(n \in {\mathbb {N}}\), and propose a characterization of \(\pi ^f\) for some special rv-functions in Hilbert spaces.
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27 January 2020
In the original publication, Example 2.2 was incorrectly published.
References
Baron, K.: On the convergence in law of iterates of random-valued functions. Aust. J. Math. Anal. Appl. 6(1), Art. 3, 9 pp. (2009)
Baron, K.: Weak law of large numbers for iterates of random-valued functions. Aequ. Math. 93, 415–423 (2019)
Baron, K., Morawiec, J.: Lipschitzian solutions to linear iterative equations revisited. Aequ. Math. 91, 161–167 (2017)
Da Prato, G.: Introduction to stochastic analysis and Malliavin calculus. Appunti. Scuola Normale Superiore di Pisa (Nuova Serie), vol 13. Edizioni della Normale, Pisa (2014)
Dudley, R.M.: Real Analysis and Probability. Cambridge Studies in Advanced Mathematics, vol. 74. Cambridge Univerity Press, Cambridge (2002)
Kapica, R.: Convergence of sequences of iterates of random-valued vector functions. Colloq. Math. 97, 1–6 (2003)
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)
Parthasarathy, K.R.: Probability Measures on Metric Spaces. Probability and Mathematical Statistics, vol. 3. Academic Press, Inc., New York (1967)
Zakusilo, O.K.: Some properties of random vectors of the form \(\sum _{n=0}^{\infty }A^i\xi _i\). Teor. Verojatnost. i Mat. Statist. (Russian) 13, 59–62 (1975)
Acknowledgements
I thank Professor Gregory Derfel for calling my attention to the paper by O. K. Zakusilo.
This research was supported by the University of Silesia Mathematics Department (Iterative Functional Equations and Real Analysis program).
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Dedicated to Professor János Aczél on his 95th birthday.
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Baron, K. Weak limit of iterates of some random-valued functions and its application. Aequat. Math. 94, 415–425 (2020). https://doi.org/10.1007/s00010-019-00650-z
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DOI: https://doi.org/10.1007/s00010-019-00650-z
Keywords
- Random-valued functions
- Iterates
- Weak limit
- Iterative equations
- Lipschitzian solutions
- Bochner integral
- Gaussian measures