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 $r\ge 1$; (3) $X$ follows a mixture of some HCM distributions and $|r|\ge 1$; (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 $r\ge 2$. 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.

24 pages, 24 figures. The statement of Theorem 3.5 is modified (weakened) because an error was found in the proof of a part of Theorem 3.5 in the published version

References in corpus (3)

• Complex analysis methods in noncommutative probability
• Classical Scale Mixtures of Boolean Stable Laws
• A constant regression characterization of a Marchenko-Pastur law