Sample Variance in Free Probability
| Autor: | Franz Lehner, Wiktor Ejsmont |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2016 |
| Předmět: |
Noncrossing partition
Probability (math.PR) 010102 general mathematics Mathematics - Operator Algebras Law of total cumulance Wigner semicircle distribution 16. Peace & justice Free probability 01 natural sciences Law of total variance Combinatorics 010104 statistics & probability 46L54 (Primary) 62E10 (Secondary) FOS: Mathematics Sample variance 0101 mathematics Operator Algebras (math.OA) Cumulant Random variable Analysis Mathematics - Probability Mathematics |
| Popis: | 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. Comment: Final version to appear in J of Funct Anal; 24 pages;Corollary 4.14 generalized; gap in the proof of Prop 4.13 fixed |
| Databáze: | OpenAIRE |
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