A Uniform Random Pointwise Ergodic Theorem

TL;DR

This paper proves that for a specific class of random increasing sequences, the modulated averages of measure-preserving systems converge to zero almost surely, extending ergodic theorems to random and modulated settings.

Contribution

It establishes a uniform random pointwise ergodic theorem for a class of random sequences with decreasing probability, a novel extension in ergodic theory.

Findings

Almost sure convergence of modulated averages to zero

Applicable to measure-preserving systems and functions orthogonal to invariant factors

Introduces uniform approximation conditions for the supremum

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.