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

Armine Bagyan () and Donald Richards ()
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Armine Bagyan: Pennsylvania State University
Donald Richards: Pennsylvania State University

Journal of Theoretical Probability, 2023, vol. 36, issue 2, 1181-1202

Abstract: Abstract We consider random walks on the cone of $$m \times m$$ 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 45:4101–4111, 2017), we obtain inequalities of Hoffmann-Jørgensen type for such random walks on the cone. In the case of the Wishart distribution $$W_m(a,I_m)$$ W m ( a , I m ) , with index parameter a and matrix parameter $$I_m$$ I m , the identity matrix, we derive explicit and computable bounds for each term appearing in the Hoffmann-Jørgensen inequalities.

Keywords: Orthogonal invariance; Riemannian metric; Submartingale; Symmetric cone; Thompson’s metric; Wishart distribution; Primary: 60E15; 62E15; Secondary: 60B20; 62E17 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10959-022-01189-7

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