An n-variate characterization of the gamma and the complex Wishart densities

Abstract

Given two independent positive random variables x and y, and the independence of xy and (1 − x) y, Tollar [1] proves that y is gamma and x is beta. He uses the involved methodology of random difference equations to prove this result. For n independent positive random variables x 2,…, x n , y, with the independence of (1 − x 2 − x 3 −…− x n ) y and ( x 2 y,…, x n y), Tollar's result [1] generalizes to the result that y is gamma and ( x 2,…, x n ) have a joint Dirichlet distribution. Similarly, given two independent p × p random positive definite symmetric matrices X and Y, with the independence of Y 1 2 XY 1 2 and Y 1 2 (I − X)Y 1 2 , it is proved that Y is Wishart and X is multivariate beta. Now given n independent p × p random symmetric positive definite matrices X 2,…, X n , Y, with the independence of (Y 1 2 X 2Y 1 2 ,…,Y 1 2 X nY 1 2 ) and Y 1 2 (I − X 2 − … − X n)Y 1 2 , we prove that Y is Wishart and ( X 2,…, X n ) have a joint multivariate Dirichlet density. We use the method of moment generating functions.