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The Lukacs-Olkin-Rubin characterization of Wishart distributions on symmetric cones. (English) Zbl 0906.62053

Summary: We characterize the Wishart distributions on a symmetric cone \(C\). If \(C= (0,+\infty)\), this has been done by E. Lukacs [Ann. Math. Stat. 26, 319–324 (1955; Zbl 0065.11103)]. If \(C\) is the cone of positive definite symmetric matrices, this has been done by I. Olkin and D. B. Rubin [ibid. 33, 1272–1280 (1962; Zbl 0111.34202)]. 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.

MSC:

62H05 Characterization and structure theory for multivariate probability distributions; copulas
60E10 Characteristic functions; other transforms
17C99 Jordan algebras (algebras, triples and pairs)
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