Weakly infinitely divisible measures on some locally compact Abelian groups

TL;DR

This paper constructs weakly infinitely divisible probability measures on certain locally compact Abelian groups, including the torus, p-adic integers, and p-adic solenoid, providing new insights and methods for these mathematical structures.

Contribution

It introduces a novel construction method for weakly infinitely divisible measures on specific Abelian groups, including a new way to construct the Haar measure on the p-adic solenoid.

Findings

Constructed weakly infinitely divisible measures on the torus, p-adic integers, and p-adic solenoid.

Provided a new construction of the Haar measure on the p-adic solenoid.

Extended the understanding of probability measures on locally compact Abelian groups.

Abstract

On the torus group, on the group of p-adic integers and on the p-adic solenoid, we give a construction of an arbitrary weakly infinitely divisible probability measure using a random element with values in a product of (possibly infinitely many) subgroups of the real numbers. As a special case of our results, we have a new construction of the Haar measure on the p-adic solenoid.