On freely quasi-infinitely divisible distributions

TL;DR

This paper introduces the class of freely quasi-infinitely divisible distributions, exploring their properties and extending classical concepts like the Bercovici-Pata bijection to include signed Lévy measures.

Contribution

It defines and studies the properties of FQID distributions, including cases with negative Gaussian parts and signed Lévy measures, and extends the Bercovici-Pata bijection to this class.

Findings

FQID distributions can have negative Gaussian components.

The total mass of the signed Lévy measure can be negative.

Extended Bercovici-Pata bijection includes classical and free characteristic triplets with signed measures.

Abstract

Inspired by the notion of quasi-infinite divisibility (QID), we introduce and study the class of freely quasi-infinitely divisible (FQID) distributions on $\mathbb{R}$, i.e. distributions which admit the free L\'{e}vy-Khintchine-type representation with signed L\'{e}vy measure. We prove several properties of the FQID class, some of them in contrast to those of the QID class. For example, a FQID distribution may have negative Gaussian part, and the total mass of its signed L\'{e}vy measure may be negative. Finally, we extend the Bercovici-Pata bijection, providing a characteristic triplet, with the L\'{e}vy measure having nonzero negative part, which is at the same time classical and free characteristic triplet.