The -transform of symmetric probability measures with unbounded supports

Authors:
Octavio Arizmendi E. and Victor Pérez-Abreu

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

MSC (2000):
Primary 46L54, 15A52

Published electronically:
February 16, 2009

MathSciNet review:
2506464

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Abstract: The Voiculescu -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 -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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Additional Information

**Octavio Arizmendi E.**

Affiliation:
School of Mathematics, University of Guanajuato, Valenciana, Guanajuato, GTO 36240, Mexico

Email:
octavius@cimat.mx

**Victor Pérez-Abreu**

Affiliation:
Research Center for Mathematics, CIMAT, Apartado Postal 402, Guanajuato, GTO 36000, Mexico

Email:
pabreu@cimat.mx

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

Keywords:
Multiplicative convolution,
free stable distribution,
random matrix

Received by editor(s):
September 29, 2008

Received by editor(s) in revised form:
November 18, 2008

Published electronically:
February 16, 2009

Additional Notes:
The first author’s research was supported by SNI-CONACYT Grant A. I. 4337 and the Statistics Laboratory of CIMAT

Communicated by:
Richard C. Bradley

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.