Regular variation and free regular infinitely divisible laws

TL;DR

This paper explores the relationship between tail behaviors of free regular infinitely divisible laws and their Lévy measures, highlighting connections to classical results and applications in random matrix theory.

Contribution

It establishes a link between tail behaviors of free regular laws and their Lévy measures, extending classical results via the Bercovici-Pata bijection.

Findings

Characterization of tail behavior in free regular laws

Connection to classical tail behavior results

Application to spectral distributions of random matrices

Abstract

In this article the relation between the tail behaviours of a free regular infinitely divisible (positively supported) probability measure and its L\'evy measure is studied. An important example of such a measure is the compound free Poisson distribution, which often occurs as a limiting spectral distribution of certain sequences of random matrices. We also describe a connection between an analogous classical result of Embrechts et al. [1979] and our result using the Bercovici-Pata bijection.