English

The normal distribution is freely selfdecomposable

Probability 2017-07-21 v2 Operator Algebras

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 μc\mu_c is freely selfdecomposable for any cc in [1,0][-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.

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}
}

Comments

22 pages

R2 v1 2026-06-22T17:39:14.046Z