The normal distribution is freely selfdecomposable
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 is freely selfdecomposable for any in . 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.
Keywords
Cite
@article{arxiv.1701.00409,
title = {The normal distribution is freely selfdecomposable},
author = {Takahiro Hasebe and Noriyoshi Sakuma and Steen Thorbjørnsen},
journal= {arXiv preprint arXiv:1701.00409},
year = {2017}
}
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22 pages