Boolean convolution of probability measures on the unit circle

TL;DR

This paper introduces the boolean convolution for probability measures on the unit circle, providing a new way to analyze the distribution of products of boolean independent unitary variables and characterizing infinitely divisible measures.

Contribution

It defines the boolean convolution on the unit circle, develops an analogue of the characteristic function, and characterizes all infinitely divisible measures under this convolution.

Findings

Defined boolean convolution for measures on the unit circle

Developed an analogue of the characteristic function for this convolution

Characterized all infinitely divisible probability measures on the circle

Abstract

We introduce the boolean convolution for probability measures on the unit circle. Roughly speaking, it describes the distribution of the product of two boolean independent unitary random variables. We find an analogue of the characteristic function and determine all infinitely divisible probability measures on the unit circle for the boolean convolution.