On free Generalized Inverse Gaussian distributions

TL;DR

This paper investigates free Generalized Inverse Gaussian distributions, establishing their key properties, relations to free Poisson variables, and their entropy-maximizing characteristics, thus extending classical GIG properties into free probability theory.

Contribution

It introduces and analyzes the properties of free GIG distributions, including infinite divisibility, unimodality, and selfdecomposability, and characterizes their relation to free Poisson variables.

Findings

fGIG is freely infinitely divisible, free regular, and unimodal.

Characterization of free GIG as solutions to certain free stochastic equations.

fGIG maximizes a free entropy functional similar to classical GIG.

Abstract

We study here properties of free Generalized Inverse Gaussian distributions (fGIG) in free probability. We show that in many cases the fGIG shares similar properties with the classical GIG distribution. In particular we prove that fGIG is freely infinitely divisible, free regular and unimodal, and moreover we determine which distributions in this class are freely selfdecomposable. In the second part of the paper we prove that for free random variables $X,Y$ where $Y$ has a free Poisson distribution one has $X\stackrel{d}{=}\frac{1}{X+Y}$ if and only if $X$ has fGIG distribution for special choice of parameters. We also point out that the free GIG distribution maximizes the same free entropy functional as the classical GIG does for the classical entropy.