Free subexponentiality

TL;DR

This paper introduces the concept of free subexponentiality in noncommutative probability, showing that distributions with regularly varying tails are included and analyzing their asymptotic tail behavior.

Contribution

It extends classical subexponentiality to free probability and characterizes distributions with regularly varying tails within this framework.

Findings

Distributions with regularly varying tails are free subexponential.

Partial sums of free random variables with such tails are tail equivalent to their maximum.

Asymptotic analysis links tail behavior to Laurent series of Cauchy and Voiculescu transforms.

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.