Free infinite divisibility for beta distributions and related ones

TL;DR

This paper investigates the free infinite divisibility of various classical distributions, establishing which are freely infinitely divisible and identifying properties related to their divisibility indicators.

Contribution

It proves free infinite divisibility for several important distributions and introduces a local property criterion to identify non-divisible cases.

Findings

Beta, beta prime, gamma, inverse gamma, Student t-, and ultraspherical distributions are freely infinitely divisible.

Gaussian, ultraspherical, and many Student t-distributions have free divisibility indicator 1.

Some distributions are not freely infinitely divisible, shown via a local property of their density functions.

Abstract

We prove that many of beta, beta prime, gamma, inverse gamma, Student t- and ultraspherical distributions are freely infinitely divisible, but some of them are not. The latter negative result follows from a local property of probability density functions. Moreover, we show that the Gaussian, ultraspherical and many of Student t-distributions have free divisibility indicator 1.