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Bessel Convolutions on Matrix Cones: Algebraic Properties and Random Walks

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  • Michael Voit

    (Universität Dortmund)

Abstract

Bessel-type convolution algebras of measures on the matrix cones of positive semidefinite q×q-matrices over ℝ,ℂ,ℍ were introduced recently by Rösler. These convolutions depend on a continuous parameter, generate commutative hypergroups, and have Bessel functions of matrix argument as characters. In this paper, we study the algebraic structure of these hypergroups. In particular, the subhypergroups, quotients, and automorphisms are classified. The algebraic properties are partially related to the properties of random walks on these matrix Bessel hypergroups. In particular, known properties of Wishart distributions, which form Gaussian convolution semigroups on these hypergroups, are put into a new light. Moreover, limit theorems for random walks are presented. In particular, we obtain strong laws of large numbers and a central limit theorem with Wishart distributions as limits.

Suggested Citation

  • Michael Voit, 2009. "Bessel Convolutions on Matrix Cones: Algebraic Properties and Random Walks," Journal of Theoretical Probability, Springer, vol. 22(3), pages 741-771, September.
    • Handle: RePEc:spr:jotpro:v:22:y:2009:i:3:d:10.1007_s10959-008-0186-2
    DOI: 10.1007/s10959-008-0186-2
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    Cited by:

    1. Michael Voit, 2017. "Dispersion and Limit Theorems for Random Walks Associated with Hypergeometric Functions of Type BC," Journal of Theoretical Probability, Springer, vol. 30(3), pages 1130-1169, September.
    2. Wojciech Matysiak, 2017. "Hypergroups and Quantum Bessel Processes of Non-integer Dimensions," Journal of Theoretical Probability, Springer, vol. 30(4), pages 1677-1691, December.

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