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arXiv · 1708.05022

A Uniform Random Pointwise Ergodic Theorem

Ben Krause·Pavel Zorin-Kranich
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Abstract

Let $a_n$ be the random increasing sequence of natural numbers which takes each value independently with decreasing probability of order $n^{-\alpha}$, $0 < \alpha < 1/2$. We prove that, almost surely, for every measure-preserving system $(X,T)$ and every $f \in L^1(X)$ orthogonal to the invariant factor, the modulated, random averages \[ \sup_{b} \Big| \frac{1}{N} \sum_{n = 1}^N b(n) T^{a_{n}} f \Big| \] converge to $0$ pointwise almost everywhere, where the supremum is taken over a set of bounded functions with certain uniform approximation properties.

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Ben Krause, Pavel Zorin-Kranich. 2017-08-16. A Uniform Random Pointwise Ergodic Theorem. https://arxiv.org/abs/1708.05022

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