Infinite convolutions of probability measures on Polish semigroups

TL;DR

This paper provides a concise introduction to the theory of infinite convolutions of probability measures on Polish semigroups, including key theorems and their proofs.

Contribution

It offers a self-contained exposition of fundamental theorems in the theory of infinite convolutions on Polish semigroups, with detailed proofs.

Findings

Proofs of Rees decomposition theorem

Proofs of Ellis–Zelazko theorem

Convolution factorization theorems

Abstract

This expository paper is intended for a short self-contained introduction to the theory of infinite convolutions of probability measures on Polish semigroups. We give the proofs of the Rees decomposition theorem of completely simple semigroups, the Ellis--\.{Z}elazko theorem, the convolution factorization theorem of convolution idempotents, and the convolution factorization theorem of cluster points of infinite convolutions.