### Abstract

Let X_{1},X_{2},…,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 χ^{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 analogue of this question and show that the only distributions, whose free sample variance is distributed according to a free χ^{2}-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.

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

Pages (from-to) | 2488-2520 |

Number of pages | 33 |

Journal | Journal of Functional Analysis |

Volume | 273 |

Issue number | 7 |

DOIs | |

Publication status | Published - 1 Oct 2017 |

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### Keywords

- Cancellation of free cumulants
- Free infinite divisibility
- Sample variance
- Wigner semicircle law

### ASJC Scopus subject areas

- Analysis

### Cite this

*Journal of Functional Analysis*,

*273*(7), 2488-2520. https://doi.org/10.1016/j.jfa.2017.05.007

**Sample variance in free probability.** / Ejsmont, Wiktor; Lehner, Franz.

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

*Journal of Functional Analysis*, vol. 273, no. 7, pp. 2488-2520. https://doi.org/10.1016/j.jfa.2017.05.007

}

TY - JOUR

T1 - Sample variance in free probability

AU - Ejsmont, Wiktor

AU - Lehner, Franz

PY - 2017/10/1

Y1 - 2017/10/1

N2 - Let X1,X2,…,Xn denote i.i.d. centered standard normal random variables, then the law of the sample variance Qn=∑i=1 n(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.

AB - Let X1,X2,…,Xn denote i.i.d. centered standard normal random variables, then the law of the sample variance Qn=∑i=1 n(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.

KW - Cancellation of free cumulants

KW - Free infinite divisibility

KW - Sample variance

KW - Wigner semicircle law

UR - http://www.scopus.com/inward/record.url?scp=85021240579&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2017.05.007

DO - 10.1016/j.jfa.2017.05.007

M3 - Article

VL - 273

SP - 2488

EP - 2520

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 7

ER -