Meixner Matrix Ensembles

Włodzimierz Bryc () and Gérard Letac
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Włodzimierz Bryc: University of Cincinnati
Gérard Letac: Université Paul Sabatier

Journal of Theoretical Probability, 2013, vol. 26, issue 1, 107-152

Abstract: Abstract We construct a family of matrix ensembles that fits Anshelevich’s regression postulates for “Meixner laws on matrices,” namely the distribution with an invariance property of X when $\mathbb{E}(\mathbf {X}^{2}|\mathbf {X}+\mathbf {Y})=a(\mathbf {X}+\mathbf {Y})^{2}+b(\mathbf {X}+\mathbf {Y})+c\mathbf {I}_{n}$ where X and Y are i.i.d. symmetric matrices of order n. We show that the Laplace transform of a general n×n Meixner matrix ensemble satisfies a system of partial differential equations which is explicitly solvable for n=2. We rely on these solutions to identify the six types of 2×2 Meixner matrix ensembles.

Keywords: Meixner laws; Random projections; Quadratic conditional moments; Matrix ensembles; Systems of PDEs; Jack polynomials; 60B20 (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1007/s10959-011-0368-1

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