Free infinite divisibility for powers of random variables

TL;DR

This paper establishes conditions under which powers of certain free random variables are freely infinitely divisible, extending classical results and providing new insights into free probability distributions.

Contribution

It identifies specific classes of free random variables whose powers are freely infinitely divisible, including free Poisson, semicircular, HCM mixtures, and beta distributions.

Findings

$X^r$ is FID for free Poisson $X$ with conditions on $r$

$|S|^r$ is FID for semicircular $S$ with $r$ in specified ranges

$S^n$ is FID for all natural numbers $n$

Abstract

We prove that $X^r$ follows an FID distribution if: (1) $X$ follows a free Poisson distribution without an atom at 0 and $r\in(-\infty,0]\cup[1,\infty)$; (2) $X$ follows a free Poisson distribution with an atom at 0 and $r\geq1$; (3) $X$ follows a mixture of some HCM distributions and $|r|\geq1$; (4) $X$ follows some beta distributions and $r$ is taken from some interval. In particular, if $S$ is a standard semicircular element then $|S|^r$ is freely infinitely divisible for $r\in(-\infty,0]\cup[2,\infty)$. Also we consider the symmetrization of the above probability measures, and in particular show that $|S|^r \,\text{sign}(S)$ is freely infinitely divisible for $r\geq2$. Therefore $S^n$ is freely infinitely divisible for every $n\in\mathbb N$. The results on free Poisson and semicircular random variables have a good correspondence with classical ID properties of powers of gamma and normal random variables.