The Matsumoto-Yor property on trees

Abstract

Viewing the Matsumoto-Yor property as a bivariate property with respect to the simple tree with two vertices and one edge, we extend it to a p-variate property with respect to any tree with p vertices. The converse of the Matsumoto-Yor property, which characterizes the product of a gamma and a generalized inverse Gaussian distribution, is extended to characterize the product of a gamma and p-1 generalized inverse Gaussian distributions. A striking feature of this characterization is that we need the independence of the components of random vectors corresponding only to the leaves of the tree. We illustrate our results with two particular trees: the two-link chain and the three-branch 'daisy'.