## Sample variance in free probability

### Wiktor Ejsmont , Franz Lehner

#### Abstract

Let X1, X2, ..., Xndenote i.i.d.centered standard normal random variables, then the law of the sample variance Qn=�ni=1(Xi−X)2is the χ2-distribution with n −1degrees of freedom. It is an open problem in classical probability to char-acterize 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 semi-circle law and more generally so-called oddlaws, by which we mean laws with vanishing higher order even cumulants. In the way of proof we derive an explicit formula for the free cumu-lants of Qnwhich shows that indeed the odd cumulants do not contribute and which exhibits an interesting connection to the concept of R-cyclicity
Author Wiktor Ejsmont (MISaF / IZM / KMiC)
Wiktor Ejsmont,,
- Katedra Matematyki i Cybernetyki
, Franz Lehner
Franz Lehner,,
-
Journal seriesJournal of Functional Analysis, ISSN 0022-1236, (A 40 pkt)
Issue year2017
Vol273
No7
Pages2488-2520
Publication size in sheets1.6
Keywords in Englishsample variance, free infinite divisibility, cancellation of free cumulants, wigner semicircle law
DOIDOI:10.1016/j.jfa.2017.05.007
URL http://dx.doi.org/10.1016/j.jfa.2017.05.007
Languageen angielski
File
 Ejsmont_Lehner_Sample_variance_in_free_ probability2017.pdf 585,37 KB
Score (nominal)40
ScoreMinisterial score = 40.0, 03-02-2019, ArticleFromJournal
Ministerial score (2013-2016) = 40.0, 03-02-2019, ArticleFromJournal
Publication indicators WoS Impact Factor: 2017 = 1.326 (2) - 2017=1.589 (5)
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