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March 2013 Free subexponentiality
Rajat Subhra Hazra, Krishanu Maulik
Ann. Probab. 41(2): 961-988 (March 2013). DOI: 10.1214/11-AOP706

Abstract

In this article, we introduce the notion of free subexponentiality, which extends the notion of subexponentiality in the classical probability setup to the noncommutative probability spaces under freeness. We show that distributions with regularly varying tails belong to the class of free subexponential distributions. This also shows that the partial sums of free random elements having distributions with regularly varying tails are tail equivalent to their maximum in the sense of Ben Arous and Voiculescu [Ann. Probab. 34 (2006) 2037–2059]. The analysis is based on the asymptotic relationship between the tail of the distribution and the real and the imaginary parts of the remainder terms in Laurent series expansion of Cauchy transform, as well as the relationship between the remainder terms in Laurent series expansions of Cauchy and Voiculescu transforms, when the distribution has regularly varying tails.

Citation

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Rajat Subhra Hazra. Krishanu Maulik. "Free subexponentiality." Ann. Probab. 41 (2) 961 - 988, March 2013. https://doi.org/10.1214/11-AOP706

Information

Published: March 2013
First available in Project Euclid: 8 March 2013

zbMATH: 1276.46054
MathSciNet: MR3077531
Digital Object Identifier: 10.1214/11-AOP706

Subjects:
Primary: 46L54
Secondary: 60G70

Keywords: Cauchy transform , convolution , Free probability , regular variation , subexponential , Voiculescu transform

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 2 • March 2013
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