The Lukacs-Olkin-Rubin characterization of Wishart distributions on symmetric cones
M. Casalis and G. Letac
Source: Ann. Statist. Volume 24, Number 2
(1996), 763-786.
Abstract
We characterize the Wishart distributions on a symmetric cone C. If $C = (0, +\infty)$, this has been done by Lukacs in 1955. If C is the cone of positive definite symmetric matrices, this has been done by Olkin and Rubin in 1962. We both shorten and extend the Olkin-Rubin proof (sometimes obscure) by using three modern ideas: (i) try to avoid artificial coordinates in differential geometry; (ii) the variance function of a natural exponential family F characterizes F; (iii) symmetric matrices are a particular example of a Euclidean simple Jordan algebra.
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Permanent link to this document: http://projecteuclid.org/euclid.aos/1032894464
Mathematical Reviews number (MathSciNet): MR1394987
Digital Object Identifier: doi:10.1214/aos/1032894464
Zentralblatt MATH identifier: 0906.62053
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