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 $X_1,\dots ,X_n$ are independent and identically distributed and the distribution of ${\sum }_{i=1}^n(X_i+a_i{)}^2$ $a_i\in \mathbb{R}$ depends only on ${\sum }_{i=1}^n{a}_i^2$ , then each $X_i$ follows the normal distribution $N(0, \sigma)$\mathrm{.}Cook1971generalizedthisresultreplacingindependenceofall$X_i$bytheindependenceof$(X_1,\dots, X_m) $\text{ and }$ (X_{m+1},\dots,X_n )$andremovingtherequirementthat$X_i$ have the same distribution. In this paper, we will give other characterizations of the normal distribution which are formulated in a similar spirit.