### Abstract

Original language | English |
---|---|

Journal | arXiv.org e-Print archive |

Publication status | Published - 22 Jul 2016 |

### Fingerprint

### Keywords

- math.OA
- math.PR
- 46L54 (Primary), 62E10 (Secondary)

### Cite this

**Sample Variance in Free Probability.** / Ejsmont, Wiktor; Lehner, Franz.

Research output: Contribution to journal › Article › Research › peer-review

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TY - JOUR

T1 - Sample Variance in Free Probability

AU - Ejsmont, Wiktor

AU - Lehner, Franz

N1 - 23 pages

PY - 2016/7/22

Y1 - 2016/7/22

N2 - 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.

AB - 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.

KW - math.OA

KW - math.PR

KW - 46L54 (Primary), 62E10 (Secondary)

M3 - Article

JO - arXiv.org e-Print archive

JF - arXiv.org e-Print archive

ER -