# 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.