Sample variance in free probability

Wiktor Ejsmont, Franz Lehner*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)2488-2520
Number of pages33
JournalJournal of Functional Analysis
Issue number7
Publication statusPublished - 1 Oct 2017


  • Cancellation of free cumulants
  • Free infinite divisibility
  • Sample variance
  • Wigner semicircle law

ASJC Scopus subject areas

  • Analysis

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