About an extension of the Matsumoto-Yor property

Abstract

If α, β > 0 are distinct and if A and B are independent non-degenerate positive random variables such that S = 1 B βA+B αA+B and T = 1 A βA+B αA+Bare independent, we prove that this happens if and only if the A and B have generalized inverse Gaussian distributions with suitable parameters. Essentially, this has already been proved in Bao and Noack ( 2021) with supplementary hypothesis on existence of smooth densities. The sources of these questions are an observation about independence properties of the exponential Brownian motion due to Matsumoto and Yor ( 2001) and a recent work of Croydon and Sasada (2000) on random recursion models rooted in the discrete Korteweg -de Vries equation, where the above result was conjectured.We also extend the direct result to random matrices proving that a matrix variate analogue of the above independence property is satisfied by independent matrix-variate GIG variables. The question of characterization of GIG random matrices through this independence property remains open.