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

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 ${\sum }_{i=1}^n{a}_i^2$ , 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 $(X_1,\dots ,X_m)\text{ and }(X_{m+1},\dots ,X_n)$ 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.

6 pages

References in corpus (3)

• Central limit theorems for sequences of multiple stochastic integrals
• On a class of free Levy laws related to a regression problem
• Cumulants in noncommutative probability II. Generalized Gaussian random variables

Cited by in corpus (1)

• A test for normality and independence based on characteristic function