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

AbstractIt is known that if the statistic Y = Σj=1n(Xj + aj)2 is drawn from a population which is distributed N(0, σ) then the distribution of Y depends on Σj=1naj2 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 Σj=1naj2, then each Xj is distributed N(0, σ). 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 Σj=1naj2, then all Xj are independent and distributed N(0, σ)