2022/03/10 by Gérard Letac, Letac, Gérard, Jacek Wesołowski +1
Mathematics · #60E05 #62E10 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2203.05404
openalex publication_date 2022/03/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
If α,β>0 are distinct and if A and B are independent non-degenerate positive random variables such that S=\tfrac1B \tfracβA+BαA+B and T=\tfrac1A \tfracβA+BαA+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.