Abstract
Let $a_{n}$ be the random increasing sequence of natural numbers which takes each value independently with probability $n^{-a}$, $0<a<1/2$, and let $p(n)=n^{1+\varepsilon}$, $0<\varepsilon<1$. We prove that, almost surely, for every measure-preserving system $(X,T)$ and every $f\in L^{1}(X)$ the modulated, random averages
\[\frac{1}{N}\sum_{n=1}^{N}e(p(n))T^{a_{n}(\omega)}f\] converge to 0 pointwise almost everywhere.
Citation
Ben Krause. Pavel Zorin-Kranich. "A random pointwise ergodic theorem with Hardy field weights." Illinois J. Math. 59 (3) 663 - 674, Fall 2015. https://doi.org/10.1215/ijm/1475266402
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