Sample variance in free probability

Let X₁,X₂,…,X_n denote i.i.d. centered standard normal random variables, then the law of the sample variance Q_n=∑_i=1 ⁿ(X_i−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 Q_n which shows that indeed the odd cumulants do not contribute and which exhibits an interesting connection to the concept of R-cyclicity.