On characterizing the normal distribution by Student's law

Abstract

Let X0,X1,…,Xn be n+1 real independent random variables (n ≧ 2) satisfying the condition P(Xk = 0) = 0 and having distributions symmetrical about zero. A necessary and sufficient condition for Xk to be identically normally distributed with zero mean, and common standard deviation a, is that Y1,Y2,…,Yn are independently distributed according to Student's law with 1,2,…,n degrees of freedom, where Y1 = Xl□1/¦X0¦,Y2 = X2□2/□(X20 + X21),…,Yn = Xn□n/□(X20l + … +X2n-1)