Limit laws for boolean convolutions

TL;DR

This paper explores the limit laws of boolean convolutions, revealing their connections to free convolutions and classical convolutions, using analytical methods from free probability theory.

Contribution

It establishes the relationship between boolean and free convolution limit laws and applies analytical tools from free probability to study their distributional behavior.

Findings

Limit laws of boolean convolutions are determined by free convolution laws.

Connections between classical and boolean convolution limit behaviors are demonstrated.

Analytical methods for free convolutions are used to analyze boolean convolutions.

Abstract

We study the distributional behavior for products, and for sums of boolean independent random variables in an infinitesimal triangular array. We show that the limit laws of boolean convolutions are determined by the limit laws of free convolutions, and vice versa. We further use these results to show several connections between the limiting distributional behavior of classical convolutions and that of boolean convolutions. The proof of our results is based on the analytical apparatus developed for free convolutions.