The Matsumoto-Yor property on trees for matrix variates of different dimensions

The paper is devoted to an extension of the multivariate Matsumoto-Yor (MY) independence property with respect to a tree with p vertices to the case where random variables corresponding to the vertices of the tree are replaced by random matrices. The converse of the p-variate MY property, which characterizes the product of one gamma and p−1 generalized inverse Gaussian distributions, is extended to characterize the product of the Wishart and p−1 matrix generalized inverse Gaussian distributions.