A characterization of the normal distribution by the independence of a pair of random vectors and a property of the noncentral chi-square statistic

It is known that if the statistic Y = [Sigma]j=1n(Xj + aj)2 is drawn from a population which is distributed N(0, [sigma]) then the distribution of Y depends on only. Kagan and Shalaevski [2] have shown that if the random variables X1, X2, ..., Xn are independent and identically distributed and the distribution of Y depends only on , then each Xj is distributed N(0, [sigma]). It is shown below that if the random vectors (X1, ..., Xm) and (Xm+1, ..., Xn) are independent and the distribution of Y depends only on , then all Xj are independent and distributed N(0, [sigma]).