A discrete-time Matsumoto-Yor theorem

TL;DR

This paper establishes a discrete-time analogue of the Matsumoto-Yor theorem by analyzing a random walk on lower triangular matrices, characterizing GIG distributions, and connecting to continuous limits.

Contribution

It introduces a discrete-time Matsumoto-Yor theorem, characterizes GIG laws via a Rogers-Pitman criterion, and links discrete and continuous identities.

Findings

Characterization of GIG laws on matrix diagonals

Discrete-time Dufresne identity proved

Recovery of Matsumoto-Yor theorem in continuous limit

Abstract

We study a random walk on the subgroup of lower triangular matrices of SL$_2$, 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.