Almost everywhere convergence of convolution powers without finite second moment

TL;DR

This paper extends a theorem on the almost everywhere convergence of convolution powers for measure-preserving transformations, removing the requirement of finite second moments for the probability measure.

Contribution

It generalizes previous results by proving convergence without assuming finite second moments of the measure.

Findings

Convolution powers converge almost everywhere under broader conditions.

The theorem applies to a wider class of probability measures.

It advances understanding of ergodic averages in dynamical systems.

Abstract

We generalize a theorem of Bellow and Calder\'on concerning the a.e. convergence of the convolution powers $\ds \mu^nf(x)=\sum_{k}\mu^n(k)f(T^k x)$ where $T$ is a measure preserving transformation of a probability space and $\mu$ is a probability measure on the integers.