On the Ritt property and weak type maximal inequalities for convolution powers on $\ell^1(\Z)$

TL;DR

This paper investigates convolution powers of probability measures on integers, focusing on measures with specific monotonicity or centered properties, and demonstrates new examples exhibiting the Ritt property and weak type maximal inequalities.

Contribution

It introduces new classes of probability measures on  with the Ritt property and establishes weak type maximal inequalities for their convolution powers.

Findings

Identified new probability measures with the Ritt property on .

Proved weak type maximal inequalities for these measures.

Extended understanding of convolution power behaviors on .

Abstract

In this paper we study the behaviour of convolution powers of probability measures $\mu$ on $\Z$, such that $(\mu(n))_{n\in \N}$ is completely monotone or such that $\nu$ is centered with a second moment. In particular we exhibit many new examples of probability measures on $\Z$ having the so called Ritt property and whose convolution powers satisfy weak type maximal inequalities in $\ell^1(\Z)$.