A Characterization of the Gaussian Distribution

Abstract

Recall that if independent random variables X 1,…, X n have the same standard normal distribution then has the chi-square distribution with n degrees of freedom. In this note, a partial reverse of this theorem is proved: if X 1,…, X n are independent infinitely divisible random variables such that has the chi-square distribution with n degrees of freedom then random variables X 1,…, X n have the same standard normal distribution.