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CRAIG-SAKAMOTO'S THEOREM FOR THE WISHART

DISTRIBUTIONS ON SYMMETRIC CONES

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G. LETAC^{1} AND H. MASSAM^{2}

^{1} *Laboratoire de Statistique et Probabilités,
Université
Paul Sabatier, 31062 Toulouse, France*

^{2} *Department of Mathematics and Statistics, York
University,*

North York, Ontario, Canada M3J 1P3
(Received May 27, 1994; revised January 30, 1995)

**Abstract.**
A version of Craig-Sakamoto's theorem says
essentially that if *X* is a *N*(0, *I*_{n}) Gaussian random variable in
R^{n}, and if *A* and *B* are (*n*, *n*) symmetric matrices, then
*X'AX* and *X'BX* (or traces of *AXX'* and *BXX'*) are independent
random variables if and only if *AB* = 0. As observed in 1951, by
Ogasawara and Takahashi, this result can be extended to the case
where *XX'* 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 algebra *E*. Similarly, we generalize
there the version of Craig's theorem given by Ogasawara and
Takahashi. We prove that if *a* and *b* are in *E* and if *W*
is Wishart distributed, then Trace *a.W* and Trace *b.W* are
independent if and only if *a.b* = 0 and *a.*(*b.x*) = *b.*(*a.x*) for
all *x* in *E*, where the . indicates Jordan product.

*Key words and phrases*:
Jordan algebra, Wishart
distributions, exponential families on convex cones.

**Source**
( TeX ,
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