Independence Characterization for Wishart and Kummer Random Matrices

Abstract

We generalize the following univariate characterization of the Kummer and Gamma distributions to the cone of symmetric positive definite matrices: let X and Y be independent, non-degenerate random variables valued in (0,∞), then U = Y /(1 +X) and V = X(1 + U) are independent if and only if X follows the Kummer distribution and Y follows the the Gamma distribution with appropriate parameters. We solve a related functional equation in the cone of symmetric positive definite matrices, which is our first main result and apply its solution to prove the characterization of Wishart and matrix-Kummer distributions, which is our second main result.