Weak law of large numbers for iterates of random-valued functions Article Swipe
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· 2018
· Open Access
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· DOI: https://doi.org/10.1007/s00010-018-0585-0
· OA: W2884895954
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 $$\int _X\psi (y)\pi (dy)$$ .
