On the free Gamma distributions

TL;DR

This paper investigates free probability analogs of Gamma distributions, proving their absolute continuity, analyzing their density properties, and exploring their asymptotic behavior as the parameter approaches zero.

Contribution

It introduces and characterizes the free Gamma distributions, including their density properties and asymptotic behavior, expanding free probability theory.

Findings

The free Gamma distributions are absolutely continuous.

Their densities are analytic and unimodal.

As the parameter approaches zero, specific asymptotic behaviors are established.

Abstract

For each positive number $\alpha$ we study the analog $\nu_alpha$ in free probability of the classical Gamma distribution with parameter $\alpha$. We prove that $\nu_\alpha$ is absolutely continuous and establish the main properties of the density, including analyticity and unimodality. We study further the asymptotic behavior of $\nu_\alpha$ as $\alpha\downarrow0$.