A discrete-time Matsumoto–Yor theorem

¹ Université Paris Cité, CNRS, MAP5, F-75006 Paris, France
² Université Gustave Eiffel, CNRS, Institut Gaspard Monge, Champs-Sur-Marne, France

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 24 November 2024
Accepted: 7 August 2025

Abstract

We study a random walk on the subgroup of lower triangular matrices of SL₂, with i.i.d. increments. We prove that the process of the lower corner of the random walk satisfies a Rogers–Pitman criterion to be a Markov chain if and only if the increments are distributed according to a Generalized Inverse Gaussian (GIG) law on their diagonals. For this, we prove a new characterization of these laws. We prove a discrete-time version of the Dufresne identity. We show how to recover the Matsumoto–Yor theorem by taking the continuous limit of the random walk.