Sample variance in free probability

Wiktor Ejsmont, Franz Lehner

Publikation: Beitrag in einer FachzeitschriftArtikelForschungBegutachtung

Abstract

Let X1,X2,…,Xn denote i.i.d. centered standard normal random variables, then the law of the sample variance Qn=∑i=1 n(Xi−X‾)2 is the χ2-distribution with n−1 degrees of freedom. It is an open problem in classical probability to characterize all distributions with this property and in particular, whether it characterizes the normal law. In this paper we present a solution of the free analogue of this question and show that the only distributions, whose free sample variance is distributed according to a free χ2-distribution, are the semicircle law and more generally so-called odd laws, by which we mean laws with vanishing higher order even cumulants. In the way of proof we derive an explicit formula for the free cumulants of Qn which shows that indeed the odd cumulants do not contribute and which exhibits an interesting connection to the concept of R-cyclicity.

Originalspracheenglisch
Seiten (von - bis)2488-2520
Seitenumfang33
FachzeitschriftJournal of Functional Analysis
Jahrgang273
Ausgabenummer7
DOIs
PublikationsstatusVeröffentlicht - 1 Okt 2017

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Free Probability
Sample variance
Cumulants
Odd
Semicircle Law
Cyclicity
Distribution-free
Explicit Formula
Open Problems
Random variable
Degree of freedom
Higher Order
Denote
Analogue

Schlagwörter

    ASJC Scopus subject areas

    • Analyse

    Dies zitieren

    Sample variance in free probability. / Ejsmont, Wiktor; Lehner, Franz.

    in: Journal of Functional Analysis, Jahrgang 273, Nr. 7, 01.10.2017, S. 2488-2520.

    Publikation: Beitrag in einer FachzeitschriftArtikelForschungBegutachtung

    Ejsmont, Wiktor ; Lehner, Franz. / Sample variance in free probability. in: Journal of Functional Analysis. 2017 ; Jahrgang 273, Nr. 7. S. 2488-2520.
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