Multiplication of free random variables and the S-transform: the case of vanishing mean

TL;DR

This paper extends the S-transform method for free multiplicative convolution to cases where the probability measures have zero mean, clarifying its applicability and interpretation in this scenario.

Contribution

It provides a new interpretation of the S-transform for measures with vanishing mean, expanding its theoretical framework.

Findings

S-transform can be correctly interpreted for zero-mean measures

The usual formula for free multiplicative convolution remains valid in this case

Clarifies the theoretical foundation for zero-mean measures in free probability

Abstract

This note extends Voiculescu's S-transform based analytical machinery for free multiplicative convolution to the case where the mean of the probability measures vanishes. We show that with the right interpretation of the S-transform in the case of vanishing mean, the usual formula makes perfectly good sense.