Hoffmann-Jørgensen Inequalities for Random Walks on the Cone of Positive Definite Matrices

2022/06/21 by Armine Bagyan, Donald Richards, Bagyan, Armine +1
Mathematics · #62E15. Secondary: 60B20 #62E17 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Point processes and geometric inequalities #Primary: 60E15 #Probability (math.PR) #Random Matrices and Applications #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.2206.10138

openalex publication_date 2022/06/21 · openalex created_date 2022/06/24 · openalex updated_date 2026/07/28

Abstract

We consider random walks on the cone of m × 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-Jørgensen 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-Jørgensen inequalities.

Related

• Random walks in cones revisited
• Matsumoto-Yor and Dufresne type theorems for a random walk on positive definite matrices
• Limit theorems for inhomogeneous random walks on GL(d,\mathbb R)
• Invariance principles for random walks in cones
• Convex hulls of random walks