Sample Variance in Free Probability

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.

Final version to appear in J of Funct Anal; 24 pages;Corollary 4.14 generalized; gap in the proof of Prop 4.13 fixed

References in corpus (5)

• On a class of free Levy laws related to a regression problem
• Cumulants in noncommutative probability II. Generalized Gaussian random variables
• Convolution, subordination and characterization problems in noncommutative probability
• On the Lukacs property for free random variables
• Dual Lukacs regressions of negative orders for non-commutative variables

Cited by in corpus (5)

• Sums of Commutators in Free Probability
• The Free Tangent Law
• Remarks on a free analogue of the beta prime distribution
• On the anti-commutator of two free random variables
• On free Generalized Inverse Gaussian distributions