Free infinite divisibility for powers of random variables

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.