The Normal Distribution Is Freely Self-decomposable

Takahiro Hasebe; Noriyoshi Sakuma; Steen Thorbjørnsen

doi:10.1093/imrn/rnx171

The Normal Distribution Is Freely Self-decomposable

Takahiro Hasebe, Noriyoshi Sakuma, Steen Thorbjørnsen^*

^*Corresponding author af dette arbejde

Institut for Matematik

Publikation: Bidrag til tidsskrift/Konferencebidrag i tidsskrift /Bidrag til avis › Tidsskriftartikel › Forskning › peer review

9 Citationer (Scopus)

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.

Originalsprog	Engelsk
Tidsskrift	International Mathematics Research Notices
Vol/bind	2019
Nummer	6
Sider (fra-til)	1758-1787
Antal sider	30
ISSN	1073-7928
DOI	https://doi.org/10.1093/imrn/rnx171
Status	Udgivet - mar. 2019

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10.1093/imrn/rnx171

https://arxiv.org/pdf/1701.00409.pdf

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Citationsformater

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AB - 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.

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The Normal Distribution Is Freely Self-decomposable

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