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

AbstractGiven 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 x2,…,xn,y, with the independence of (1 − x2 − x3 −…− xn)y and (x2y,…,xny), Tollar's result [1] generalizes to the result that y is gamma and (x2,…,xn) have a joint Dirichlet distribution.Similarly, given two independent p × p random positive definite symmetric matrices X and Y, with the independence of Y12XY12 and Y12(I − X)Y12, it is proved that Y is Wishart and X is multivariate beta. Now given n independent p × p random symmetric positive definite matrices X2,…,Xn,Y, with the independence of (Y12X2Y12,…,Y12XnY12) and Y12(I − X2 − … − Xn)Y12, we prove that Y is Wishart and (X2,…,Xn) have a joint multivariate Dirichlet density. We use the method of moment generating functions