A Characterization of the Dirichlet Distribution

Abstract Let X1, X2, ···, Xk be positive random variables such that Σi = 1 k Xi < 1. It is shown, under the assumption of continuous pdf's, that if is independent of the set {Xji j ≠ i} for every i= 1, 2, ···, k then X1, X2, ···, Xk have a Dirichlet distribution, namely αi positive, xi ≥ 0, Σk i = 1 xi < 1.