Bessel Convolutions on Matrix Cones: Algebraic Properties and Random Walks

Michael Voit ()
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Michael Voit: Universität Dortmund

Journal of Theoretical Probability, 2009, vol. 22, issue 3, 741-771

Abstract: 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.

Keywords: Bessel functions of matrix argument; Product formula; Hypergroups; Automorphisms; Subhypergroups; Wishart distributions; Random walks on matrix cones; Central limit theorem; Strong laws of large numbers; 43A62; 33C52; 43A90; 60B15; 60F05 (search for similar items in EconPapers)
Date: 2009
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s10959-008-0186-2

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