Abstract
A version of Craig-Sakamoto's theorem says essentially that ifX is aN(O,I n ) Gaussian random variable in ∝n, and ifA andB are (n, n) symmetric matrices, thenX′AX andX′BX (or traces ofAXX′ andBXX′) are independent random variables if and only ifAB=0. As observed in 1951, by Ogasawara and Takahashi, this result can be extended to the case whereXX′ is replaced by a Wishart random variable. Many properties of the ordinary Wishart distributions have recently been extended to the Wishart distributions on the symmetric cone generated by a Euclidean Jordan algebraE. Similarly, we generalize there the version of Craig's theorem given by Ogasawara and Takahashi. We prove that ifa andb are inE and ifW is Wishart distributed, then Tracea.W and Traceb.W are independent if and only ifa.b=0 anda.(b.x)=b.(a.x) for allx inE, where the. indicates Jordan product.
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Partially supported by NATO grant 92.13.47.
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Letac, G., Massam, H. Craig-Sakamoto's theorem for the Wishart distributions on symmetric cones. Ann Inst Stat Math 47, 785–799 (1995). https://doi.org/10.1007/BF01856547
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DOI: https://doi.org/10.1007/BF01856547