Isotropy and Sphericity: Some Characterisations of the Normal Distribution
Letac, Gerard
Ann. Statist., Tome 9 (1981) no. 1, p. 408-417 / Harvested from Project Euclid
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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.
Publié le : 1981-03-14
Classification:  Normal distribution,  Cauchy distribution,  cylindrical-distribution,  62E10,  60B15 
@article{1176345406,
     author = {Letac, Gerard},
     title = {Isotropy and Sphericity: Some Characterisations of the Normal Distribution},
     journal = {Ann. Statist.},
     volume = {9},
     number = {1},
     year = {1981},
     pages = { 408-417},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1176345406}
}

Letac, Gerard. Isotropy and Sphericity: Some Characterisations of the Normal Distribution. Ann. Statist., Tome 9 (1981) no. 1, pp.  408-417. http://gdmltest.u-ga.fr/item/1176345406/