The normal distribution is $\boxplus$-infinitely divisible

TL;DR

This paper proves that the classical normal distribution is infinitely divisible under free additive convolution, expanding understanding of distribution properties in free probability theory.

Contribution

It demonstrates the free infinite divisibility of the normal distribution and a family of Askey-Wimp-Kerov distributions, including the normal, as a new class of such distributions.

Findings

Normal distribution is freely infinitely divisible.

Askey-Wimp-Kerov distributions are freely infinitely divisible.

Normal distribution is one of few distributions infinitely divisible under both classical and free convolutions.

Abstract

We prove that the classical normal distribution is infinitely divisible with respect to the free additive convolution. We study the Voiculescu transform first by giving a survey of its combinatorial implications and then analytically, including a proof of free infinite divisibility. In fact we prove that a subfamily Askey-Wimp-Kerov distributions are freely infinitely divisible, of which the normal distribution is a special case. At the time of this writing this is only the third example known to us of a nontrivial distribution that is infinitely divisible with respect to both classical and free convolution, the others being the Cauchy distribution and the free 1/2-stable distribution.