## The Annals of Probability

### Free subexponentiality

#### 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.

#### Article information

Source
Ann. Probab., Volume 41, Number 2 (2013), 961-988.

Dates
First available in Project Euclid: 8 March 2013

https://projecteuclid.org/euclid.aop/1362750947

Digital Object Identifier
doi:10.1214/11-AOP706

Mathematical Reviews number (MathSciNet)
MR3077531

Zentralblatt MATH identifier
1276.46054

Subjects
Primary: 46L54: Free probability and free operator algebras
Secondary: 60G70: Extreme value theory; extremal processes

#### Citation

Hazra, Rajat Subhra; Maulik, Krishanu. Free subexponentiality. Ann. Probab. 41 (2013), no. 2, 961--988. doi:10.1214/11-AOP706. https://projecteuclid.org/euclid.aop/1362750947

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