The 𝑆-transform of symmetric probability measures with unbounded supports

Arizmendi E., Octavio; Pérez-Abreu, Victor

doi:10.1090/S0002-9939-09-09841-4

The $S$-transform of symmetric probability measures with unbounded supports
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by Octavio Arizmendi E. and Victor Pérez-Abreu

Proc. Amer. Math. Soc. 137 (2009), 3057-3066

DOI: https://doi.org/10.1090/S0002-9939-09-09841-4

Published electronically: February 16, 2009

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Abstract:

The Voiculescu $S$-transform is an analytic tool for computing free multiplicative convolutions of probability measures. It has been studied for probability measures with non-negative support and for probability measures having all moments and zero mean. We extend the $S$-transform to symmetric probability measures with unbounded support and without moments. As an application, a representation of symmetric free stable measures is derived as a multiplicative convolution of the semicircle measure with a positive free stable measure.

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