Isotropy and Sphericity: Some Characterisations of the Normal Distribution

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.