Hoffmann-J{\o}rgensen Inequalities for Random Walks on the Cone of Positive Definite Matrices

TL;DR

This paper extends Hoffmann-J{\

Contribution

It develops Hoffmann-J{\

Findings

Provides inequalities for random walks on positive definite matrices

Derives explicit bounds for Wishart distribution cases

Utilizes orthogonally invariant distributions and Riemannian metrics

Abstract

We consider random walks on the cone of $m \times m$ positive definite matrices, where the underlying random matrices have orthogonally invariant distributions on the cone and the Riemannian metric is the measure of distance on the cone. By applying results of Khare and Rajaratnam (Ann. Probab., 45 (2017), 4101--4111), we obtain inequalities of Hoffmann-J{\o}rgensen type for such random walks on the cone. In the case of the Wishart distribution $W_m(a,I_m)$, with index parameter $a$ and matrix parameter $I_m$, the identity matrix, we derive explicit and computable bounds for each term appearing in the Hoffmann-J{\o}rgensen inequalities.