Multivariate Matsumoto–Yor property is rather restrictive

Abstract

Matsumoto and Yor [2001. An analogue of Pitman's 2 M - X theorem for exponential Wiener functionals. Part II: the role of the GIG laws. Nagoya Math. J. 162, 65–86] discovered an interesting invariance property of a product of the generalized inverse Gaussian (GIG) and the gamma distributions. For univariate random variables or symmetric positive definite random matrices it is a characteristic property for this pair of distributions. It appears that for random vectors the Matsumoto–Yor property characterizes only very special families of multivariate GIG and gamma distributions: components of the respective random vectors are grouped into independent subvectors, each subvector having linearly dependent components. This complements the version of the multivariate Matsumoto–Yor property on trees and related characterization obtained in Massam and Wesołowski [2004. The Matsumoto–Yor property on trees. Bernoulli 10, 685–700].