Harmonic analysis for the bi-free partial S-transform

TL;DR

This paper advances bi-free harmonic analysis by examining the bi-free partial S-transform, enabling new insights into limit theorems and infinite divisibility in the context of multiplicative bi-free convolution.

Contribution

It introduces a detailed analysis of the bi-free partial S-transform and applies it to study limit theorems and infinite divisibility, extending previous bi-free harmonic analysis work.

Findings

Characterization of the bi-free partial S-transform

Results on limit theorems in bi-free probability

Conditions for infinite divisibility in bi-free convolution

Abstract

This is a continuation of our previous work in bi-free harmonic analysis for commuting left and right variables. Here we analyze the bi-free partial S-transform and use the results to study limit theorems and infinite divisibility relative to the multiplicative bi-free convolution.