Free infinite divisibility for generalized power distributions with free Poisson term

TL;DR

This paper investigates conditions under which certain generalized power distributions with free Poisson components are freely infinitely divisible, using complex analysis and free cumulant calculations to establish new results.

Contribution

It provides new criteria for free infinite divisibility of generalized power distributions with free Poisson terms, including specific parameter ranges for various distributions.

Findings

Free infinite divisibility holds for free GIG distributions when |r|≥1.

$(S+u)^r$ is FID for semicircle law when r≤-1 and u≥2.

Beta distributions' powers are FID under certain parameter and exponent conditions.

Abstract

We study free infinite divisibility (FID) for a class which is called generalized power distributions with free Poisson term by using a complex analytic technique and a calculation for the free cumulants and Hankel determinants. In particular, our main result implies that (i) if $X$ follows the free Generalized Inverse Gaussian distribution, then $X^r$ follows an FID distribution when $|r|\ge1$, (ii) if $S$ follows the standard semicircle law and $u\ge 2$, then $(S+u)^r$ follows an FID distribution when $r\le -1$, and (iii) if $B_p$ follows the beta distribution with parameters $p$ and $3/2$, then (a) $B_p^r$ follows an FID distribution when $|r|\ge 1$ and $0<p\le 1/2$, and (b) $B_p^r$ follows an FID distribution when $r\le -1$ and $p>1/2$.