Sample Variance in Free Probability

TL;DR

This paper characterizes distributions with free sample variance having a free chi-squared distribution, showing they are limited to semicircle and odd laws, thus solving a free probability analog of a classical problem.

Contribution

It provides a complete characterization of distributions with free sample variance following a free chi-squared distribution, identifying semicircle and odd laws as the only solutions.

Findings

Only semicircle and odd laws have free sample variance with free chi-squared distribution.

Derived explicit formula for free cumulants of the free sample variance.

Connected free cumulants to the concept of R-cyclicity.

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.