The normal distribution is freely selfdecomposable

TL;DR

This paper proves that classical normal distributions are freely selfdecomposable and extends this property to a family of distributions, enriching the understanding of free probability theory.

Contribution

It establishes that normal distributions are freely selfdecomposable and characterizes this property for the Askey-Wimp-Kerov distributions within a specific parameter range.

Findings

Normal distributions are freely selfdecomposable.

Askey-Wimp-Kerov distributions with c in [-1,0] are freely selfdecomposable.

Provides a characterization of freely selfdecomposable distributions via free cumulant transforms.

Abstract

The class of selfdecomposable distributions in free probability theory was introduced by Barndorff-Nielsen and the third named author. It constitutes a fairly large subclass of the freely infinitely divisible distributions, but so far specific examples have been limited to Wigner's semicircle distributions, the free stable distributions, two kinds of free gamma distributions and a few other examples. In this paper, we prove that the (classical) normal distributions are freely selfdecomposable. More generally it is established that the Askey-Wimp-Kerov distribution $\mu_c$ is freely selfdecomposable for any $c$ in $[-1,0]$. The main ingredient in the proof is a general characterization of the freely selfdecomposable distributions in terms of the derivative of their free cumulant transform.