A discrete-time Matsumoto-Yor theorem
Probability
2024-09-04 v1
Abstract
We study a random walk on the subgroup of lower triangular matrices of SL, 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.
Cite
@article{arxiv.2409.01044,
title = {A discrete-time Matsumoto-Yor theorem},
author = {Charlie Herent},
journal= {arXiv preprint arXiv:2409.01044},
year = {2024}
}