---
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.'
author:
- |
    Takahiro Hasebe\
    Department of Mathematics,\
    Hokkaido University\
    thasebe@math.sci.hokudai.ac.jp
- |
    Kamil Szpojankowski\
    Faculty of Mathematics and Information Science\
    Warsaw University of Technology\
    k.szpojankowski@mini.pw.edu.pl
title: On free generalized inverse gaussian distributions
---

Introduction
============

Free probability was introduced by Voiculescu in [@Voi85] as a non-commutative probability theory where one defines a new notion of independence, so called freeness or free independence. Non-commutative probability is a counterpart of the classical probability theory where one allows random variables to be 