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A Characterisation of the Gaussian Distribution through the Sample Variance

Abstract

A classical result states that the sample variance of a standard Gaussian sample has the chi-square distribution. In this note, a partial reverse of this result is proved for independent infinitely divisible random variables X 1,…,X n ,n≥2. If n≥3,\(\mathbb {E}X_{1}=\cdots =\mathbb {E}X_{n}\) and the random variable \(n{S^{2}_{n}}=(X_{1}-\overline {X})^{2}+\cdots +(X_{n}-\overline {X})^{2}\) where \(\overline {X}=(X_{1}+\cdots +X_{n})/n,\) has the chi-square distribution with n−1 degrees of freedom then X 1,…,X n are Gaussian random variables with \(\mathbb {E}(X_{1}-\mathbb {E}X_{1})^{2}=\cdots =\mathbb {E}(X_{n}-\mathbb {E}X_{n})^{2}=1\). In the case n=2, the random variable \(2{S^{2}_{2}}\) has the chi-square distribution with 1 degree of freedom if and only if X 1 and X 2 are Gaussian random variables with \(\mathbb {E}X_{1}=\mathbb {E}X_{2}\) and \(\mathbb {E}(X_{1}-\mathbb {E}X_{1})^{2}+\mathbb {E}(X_{2}-\mathbb {E}X_{2})^{2}=2\).

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Correspondence to Victor M. Kruglov.

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Research supported by the Russian Scientific Foundation, project 14-11-00362.

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Golikova, N.N., Kruglov, V.M. A Characterisation of the Gaussian Distribution through the Sample Variance. Sankhya A 77, 330–336 (2015). https://doi.org/10.1007/s13171-014-0060-5

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  • DOI: https://doi.org/10.1007/s13171-014-0060-5

Keywords and phrases

  • Gaussian distribution
  • chi-squared distribution
  • distribution theory

AMS 2000 subject classifications

  • Primary 60E07
  • Secondary 62E10