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Transactions of the American Mathematical Society

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On Wishart and noncentral Wishart distributions on symmetric cones


Author: Eberhard Mayerhofer
Journal: Trans. Amer. Math. Soc. 371 (2019), 7093-7109
MSC (2010): Primary 60B15; Secondary 05A10, 05A05, 33C80
DOI: https://doi.org/10.1090/tran/7754
Published electronically: January 16, 2019
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Abstract: Necessary conditions for the existence of noncentral Wishart distributions are given. Our method relies on positivity properties of spherical polynomials on Euclidean Jordan algebras and advances an approach by Peddada and Richards [Ann. Probab. 19 (1991), pp. 868-874] where only a special case (positive semidefinite matrices, rank $ 1$ noncentrality parameter) is treated. Not only do the shape parameters need to be in the Wallach set--as is the case for Riesz measures--but also the rank of the noncentrality parameter is constrained by the size of the shape parameter. This rank condition has recently been proved with different methods for the special case of symmetric, positive semidefinite matrices (Letac and Massam; Graczyk, Malecki, and Mayerhofer).


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Additional Information

Eberhard Mayerhofer
Affiliation: Department of Mathematics and Statistics, University of Limerick, Castletroy, Ireland
Email: eberhard.mayerhofer@ul.ie

DOI: https://doi.org/10.1090/tran/7754
Keywords: Wishart distribution, Jack polynomials, generalized binomial coefficients, symmetric cones
Received by editor(s): October 23, 2017
Received by editor(s) in revised form: February 20, 2018
Published electronically: January 16, 2019
Article copyright: © Copyright 2019 American Mathematical Society