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A Characterization of a Matric-Variate Exponential Distribution

Summary 
 
Suppose S is a positive definite m x m random matrix and S> Ω denotes the event that S—Ω is positive definite, Ω being a constant positive definite matrix. Under very mild regularity conditions, we show that the constraint Pr(S>Ω1+Ω2 | S>Ω2) = Pr(S>Ω1) implies that S has a Wishart distribution on m+ 1 degrees of freedom.