# Sample Variance in Free Probability

Wiktor Ejsmont, Franz Lehner

Research output: Contribution to journalArticleResearchpeer-review

### Abstract

Let $X_1, X_2,\dots, X_n$ denote i.i.d.~centered standard normal random variables, then the law of the sample variance $Q_n=\sum_{i=1}^n(X_i-\bar{X})^2$ is the $\chi^2$-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 analog of this question and show that the only distributions, whose free sample variance is distributed according to a free $\chi^2$-distribution, are the semicircle law and more generally so-called \emph{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 $Q_n$ which shows that indeed the odd cumulants do not contribute and which exhibits an interesting connection to the concept of $R$-cyclicity.
Original language English arXiv.org e-Print archive Published - 22 Jul 2016

### Fingerprint

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

### Keywords

• math.OA
• math.PR
• 46L54 (Primary), 62E10 (Secondary)

### Cite this

Sample Variance in Free Probability. / Ejsmont, Wiktor; Lehner, Franz.

In: arXiv.org e-Print archive, 22.07.2016.

Research output: Contribution to journalArticleResearchpeer-review

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