On characterizing the normal distribution by Student's law

Abstract

Let X 0 ,X 1 ,…,X n be n+1 real independent random variables (n ≧ 2) satisfying the condition P(X k = 0) = 0 and having distributions symmetrical about zero. A necessary and sufficient condition for X k to be identically normally distributed with zero mean, and common standard deviation a, is that Y 1 ,Y 2 ,…,Y n are independently distributed according to Student's law with 1,2,…,n degrees of freedom, where Y 1 = X l □1/¦X 0 ¦,Y 2 = X 2 □2/□(X 2 0 + X 2 1 ),…,Y n = X n □n/□(X 2 0 l + … +X 2 n-1 )