Products of free random variables and k-divisible partitions

TL;DR

This paper develops a combinatorial framework for understanding the moments and free cumulants of products of free random variables, leading to new bounds and generalizations in free probability theory.

Contribution

It introduces a formula linking moments and free cumulants to k-divisible partitions, generalizing previous results and removing the need for identical measures.

Findings

Support of free multiplicative convolution grows at most linearly with k

Provides a new proof for bounds on the support of $oxtimes k$ convolution

Generalizes Kargin's results to non-identical measures

Abstract

We derive a formula for the moments and the free cumulants of the multiplication of $k$ free random variables in terms of $k$-equal and $k$-divisible non-crossing partitions. This leads to a new simple proof for the bounds of the right-edge of the support of the free multiplicative convolution $\mu^{\boxtimes k}$, given by Kargin which show that the support grows at most linearly with $k$. Moreover, this combinatorial approach generalize the results of Kargin since we do not require the convolved measures to be identical. We also give further applications, such as a new proof of the limit theorem of Sakuma and Yoshida.