The Matsumoto--Yor Property and Its Converse on Symmetric Cones

TL;DR

This paper extends the Matsumoto--Yor property to symmetric cones, proving a converse result with minimal smoothness assumptions by solving a related functional equation, thus broadening the theoretical understanding of distributional properties in multivariate analysis.

Contribution

It proves the converse of the symmetric cone-variate Matsumoto--Yor property, generalizing previous results and reducing smoothness assumptions through a new functional equation solution.

Findings

Extended the Matsumoto--Yor property to symmetric cones.

Proved the converse of the property with minimal smoothness assumptions.

Solved a new functional equation related to symmetric cones.

Abstract

The Matsumoto--Yor (MY) property of the generalized inverse Gaussian and gamma distributions has many generalizations. As it was observed in (Letac and Weso{\l}owski in Ann Probab 28:1371--1383, 2000) the natural framework for the multivariate MY property is symmetric cones; however they prove their results for the cone of symmetric positive definite real matrices only. In this paper, we prove the converse to the symmetric cone-variate MY property, which extends some earlier results. The smoothness assumption for the densities of respective variables is reduced to the continuity only. This enhancement was possible due to the new solution of a related functional equation for real functions defined on symmetric cones.