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

Author

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  • Armine Bagyan

    (Pennsylvania State University)

  • Donald Richards

    (Pennsylvania State University)

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.

Suggested Citation

  • Armine Bagyan & Donald Richards, 2023. "Hoffmann-Jørgensen Inequalities for Random Walks on the Cone of Positive Definite Matrices," Journal of Theoretical Probability, Springer, vol. 36(2), pages 1181-1202, June.
    • Handle: RePEc:spr:jotpro:v:36:y:2023:i:2:d:10.1007_s10959-022-01189-7
    DOI: 10.1007/s10959-022-01189-7
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    References listed on IDEAS

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    1. Manabu Asai & Michael McAleer & Jun Yu, 2006. "Multivariate Stochastic Volatility," Microeconomics Working Papers 22058, East Asian Bureau of Economic Research.
    2. Manabu Asai & Michael McAleer & Jun Yu, 2006. "Multivariate Stochastic Volatility: A Review," Econometric Reviews, Taylor & Francis Journals, vol. 25(2-3), pages 145-175.
    3. Krishnaiah, P. R. & Chang, T. C., 1971. "On the exact distributions of the extreme roots of the Wishart and MANOVA matrices," Journal of Multivariate Analysis, Elsevier, vol. 1(1), pages 108-117, April.
    4. Michael Browne, 1968. "A comparison of factor analytic techniques," Psychometrika, Springer;The Psychometric Society, vol. 33(3), pages 267-334, September.
    5. Elena Hadjicosta & Donald Richards, 2020. "Integral transform methods in goodness-of-fit testing, II: the Wishart distributions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(6), pages 1317-1370, December.
    6. Elena Hadjicosta & Donald Richards, 2020. "Integral transform methods in goodness-of-fit testing, I: the gamma distributions," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 83(7), pages 733-777, October.
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    Cited by:

    1. Apoorva Khare, 2024. "Probability inequalities for strongly left-invariant metric semigroups/monoids, including all lie groups," Indian Journal of Pure and Applied Mathematics, Springer, vol. 55(3), pages 1026-1039, September.

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