# Using Boolean cumulants to study multiplication and anticommutators of free random variables

## TL;DR

This paper explores how Boolean cumulants can be employed to analyze the multiplication and anticommutators of freely independent random variables, providing new insights into their distributions.

## Contribution

It introduces a novel application of Boolean cumulants to study the $*$-distribution of products and anticommutators of free random variables.

## Key findings

- Boolean cumulants effectively characterize the distributions of products and anticommutators.
- New formulas relating Boolean cumulants to free multiplicative and anticommutator operations.
- Enhanced understanding of the $*$-distribution in free probability theory.

## Abstract

We study how Boolean cumulants can be used in order to address operations with freely independent random variables, particularly in connection to the $*$-distribution of the product of two selfadjoint freely independent random variables, and in connection to the distribution of the anticommutator of such random variables.