The Normal Distribution Is Freely Self-decomposable

Takahiro Hasebe, Noriyoshi Sakuma, Steen Thorbjørnsen*

*Corresponding author for this work

Research output: Contribution to journal/Conference contribution in journal/Contribution to newspaperJournal articleResearchpeer-review

Abstract

The class of self-decomposable distributions in free probability theory was introduced by Barndorff-Nielsen and Thorbjørnsen. 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 article, we prove that the (classical) normal distributions are freely self-decomposable. More generally it is established that the Askey-Wimp-Kerov distribution μc is freely self-decomposable for any c in [-1, 0]. The main ingredient in the proof is a general characterization of the freely self-decomposable distributions in terms of the derivative of their free cumulant transform.

Original languageEnglish
JournalInternational Mathematics Research Notices
Volume2019
Issue6
Pages (from-to)1758-1787
Number of pages30
ISSN1073-7928
DOIs
Publication statusPublished - Mar 2019

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