The normal distribution is $\boxplus$-infinitely divisible
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.
Keywords
Cite
@article{arxiv.0910.4263,
title = {The normal distribution is $\boxplus$-infinitely divisible},
author = {Serban T. Belinschi and Marek Bozejko and Franz Lehner and Roland Speicher},
journal= {arXiv preprint arXiv:0910.4263},
year = {2012}
}
Comments
AMS LaTeX, 29 pages, using tikz and 3 eps figures; new proof including infinite divisibility of certain Askey-Wilson-Kerov distibutions