## The Annals of Statistics

### Isotropy and Sphericity: Some Characterisations of the Normal Distribution

Gerard Letac

#### Abstract

Main result: $X_1, X_2, \cdots, X_n$ are independent random variables valued in Euclidean spaces $E_1, E_2, \cdots, E_n$ such that $P\lbrack X_j = 0 \rbrack = 0$ for all $j$. Denote $R = \lbrack \sum^n_{j = 1} \|X_j\|^2 \rbrack^{1/2}$. Suppose that $(R^{-1}X_1, R^{-1}X_2, \cdots, R^{-1}X_n)$ is uniformly distributed on the sphere of $\oplus^n_{j = 1} E_j$. Then the $X_j$ are normal if $n \geq 3$. The case $n = 2$ and the case of Hilbert spaces are also studied.

#### Article information

Source
Ann. Statist., Volume 9, Number 2 (1981), 408-417.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176345406

Digital Object Identifier
doi:10.1214/aos/1176345406

Mathematical Reviews number (MathSciNet)
MR606624

Zentralblatt MATH identifier
0462.62014

JSTOR