About an extension of the Matsumoto-Yor property

TL;DR

This paper characterizes when certain transformations of independent positive random variables are independent, showing they must follow generalized inverse Gaussian distributions, and extends these results to matrix-variate cases.

Contribution

It proves a characterization of GIG distributions via independence properties and extends the result to matrix-variate GIG variables, building on prior conjectures.

Findings

Independence of transformed variables implies GIG distribution.

Extension of the property to matrix-variate GIG variables.

Open question on characterizing GIG matrices through independence.

Abstract

If $\alpha,\beta>0$ are distinct and if $A$ and $B$ are independent non-degenerate positive random variables such that $$S=\tfrac{1}{B}\,\tfrac{\beta A+B}{\alpha A+B}\quad \mbox{and}\quad T=\tfrac{1}{A}\,\tfrac{\beta A+B}{\alpha A+B} $$ are 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.