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 and a \( {\mathcal {B}} \otimes {\mathcal {A}} \)-measurable \( f: X \times \Omega \rightarrow X \) we consider its iterates \( f^n\) 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 provide a simple criterion for the existence of a probability Borel measure \(\pi \) on X such that for every \( x \in X \) and for every Lipschitz and bounded \(\psi :X \rightarrow {\mathbb {R}}\) the sequence \(\left( \frac{1}{n}\sum _{k=0}^{n-1} \psi \big (f^k(x,\cdot )\big )\right) _{n \in {\mathbb {N}}}\) converges in probability to \(\int _X\psi (y)\pi (dy)\).
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Acknowledgements
I thank Professor Rafał Kapica for calling my attention to the problem.
This research was supported by the University of Silesia Mathematics Department (Iterative Functional Equations and Real Analysis program).
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To the memory of Professor Marek Kuczma and Professor Győrgy Targoński.
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Baron, K. Weak law of large numbers for iterates of random-valued functions. Aequat. Math. 93, 415–423 (2019). https://doi.org/10.1007/s00010-018-0585-0
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DOI: https://doi.org/10.1007/s00010-018-0585-0