Randomly Perturbed Ergodic Averages

TL;DR

This paper establishes universal pointwise convergence and variational inequalities for random ergodic averages along randomly perturbed times, extending previous results to broader classes of functions in $L^2$ with specific integrability conditions.

Contribution

It proves universal convergence results for a class of random averages with minimal smoothness assumptions, generalizing prior work on ergodic averages along perturbed times.

Findings

Proves universal pointwise convergence for $L^2$ functions with certain integrability.

Establishes variational inequalities for averages with smoothing properties.

Extends convergence results to broader classes of functions and perturbations.

Abstract

Convergence properties of random ergodic averages have been extensively studied in the literature. In these notes, we exploit a uniform estimate by Cohen \& Cuny who showed convergence of a series along randomly perturbed times for functions in $L^2$ with $\int \max(1,\log (1+|t|)) d\mu_f<\infty$. We prove universal pointwise convergence of a class of random averages along randomly perturbed times for $L^2$ functions with $\int \max(1,\log\log(1+|t|)) d\mu_f<\infty$. For averages with additional smoothing properties, we obtain a universal variational inequality as well as universal pointwise convergence of a series define by them for all functions in $L^2$.