A characterization of the normal distribution by the independence of a pair of random vectors

Kagan and Shalaevski (1967) have shown that if the random variables X1,…,XnX1,…,Xn are i.i.d. and the distribution of ∑i=1n(Xi+ai)2ai∈Rai∈R depends only on ∑i=1nai2, then each Xi∼N(0,σ)Xi∼N(0,σ). In this paper, we will give other characterizations of the normal distribution which are formulated in a similar spirit.