Matsumoto–Yor and Dufresne type theorems for a random walk on positive definite matrices

Arista, Jonas and Bisi, Elia and O’Connell, Neil (2024) Matsumoto–Yor and Dufresne type theorems for a random walk on positive definite matrices. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 60 (2). pp. 923-945. ISSN 0246-0203

Abstract

We establish analogues of the geometric Pitman 2M - X theorem of Matsumoto and Yor and of the classical Dufresne identity, for a multiplicative random walk on positive definite matrices with Beta type II distributed increments. The Dufresne type identity provides another example of a stochastic matrix recursion, as considered by Chamayou and Letac (J. Theoret. Probab. 12, 1999), that admits an explicit solution.