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

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