Cauchy-Stieltjes kernels families and free multiplicative convolution

TL;DR

This paper explores how free multiplicative convolution impacts the pseudo-variance function of Cauchy-Stieltjes kernel families, establishing limit theorems involving various types of free and boolean convolutions.

Contribution

It introduces new insights into the effect of free multiplicative convolution on Cauchy-Stieltjes kernel families and develops related limit theorems using variance function techniques.

Findings

Effect of free multiplicative convolution on pseudo-variance functions clarified

Limit theorems established involving free and boolean additive convolutions

Variance function machinery applied to convolution analysis

Abstract

In this paper, we determine the effect of the free multiplicative convolution on the pseudo-variance function of a Cauchy-Stieltjes kernel family. We then use the machinery of variance functions to establish some limit theorems related to this type of convolution and involving the free additive convolution and the boolean additive convolution.