Abstract
Let X1,X2,…,Xn denote i.i.d. centered standard normal random variables, then the law of the sample variance Qn=∑i=1n(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.
Highlights
Many questions in classical statistics involve characterization problems, which usually are instances of the following very general question: Problem 1.1
In the present paper we solve the free version of the following problem, which is still open in classical probability and might be called χ2-conjecture, see [17, p. 466]: Conjecture 1.2
In particular we show that sample variance preserves free infinite divisibility
Summary
Many questions in classical statistics involve characterization problems, which usually are instances of the following very general question: Problem 1.1. Bercovici and Voiculescu [3] showed that there exist free random variables with a finite number of nonvanishing free cumulants which are not semicircular, see [7] for a characterization of such distributions This class of distributions appears in some (but not all) free characterization problems which are analogues of classical characterizations of the normal law, cf [21, 7]. In the present paper we show that Conjecture 1.2 falls in this class of problems and instead of Wigner laws we obtain the class of odd laws, i.e., laws with vanishing even cumulants Such laws do not exist in classical probability, but can be constructed in free probability using the results of [7].
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