A Characterization of the Normal Distribution by the Independence of a Pair of Random Vectors

TL;DR

This paper explores new characterizations of the normal distribution based on the independence of specific pairs of random vectors, extending previous results by Kagan, Shalaevski, and Cook.

Contribution

It introduces novel characterizations of the normal distribution involving independence conditions of random vectors, generalizing earlier theorems.

Findings

New characterizations of the normal distribution established

Independence of specific pairs of vectors implies normality

Extends previous results by relaxing assumptions

Abstract

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 $\sum_{i=1}^na_i^2$ , then each $X_i$ follows the normal distribution $N(0, \sigma)$. Cook 1971 generalized this result replacing independence of all $X_i$ by the independence of $(X_1,\dots, X_m) \textrm{ and } (X_{m+1},\dots,X_n )$ and removing the requirement that $X_i$ have the same distribution. In this paper, we will give other characterizations of the normal distribution which are formulated in a similar spirit.