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The Wishart Distribution on Symmetric Cones

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Non-commutative and Non-associative Algebra and Analysis Structures (SPAS 2019)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 426))

Abstract

In  this paper we discuss the extension of the Wishart probability distributions in higher dimension based on the boundary points of the symmetric cones in Jordan algebras. The symmetric cones form a basis for the construction of the degenerate and non-degenerate Wishart distributions in the field of \({{\,\textrm{Herm}\,}}(m,\mathbb {C})\), \({{\,\textrm{Herm}\,}}(m,\mathbb {H})\), \({{\,\textrm{Herm}\,}}(3,\mathbb {O})\) that denotes respectively the Jordan algebra of all Hermitian matrices of size \(m\times m\) with complex entries, the skew field \(\mathbb {H}\) of quaternions, and the algebra \(\mathbb {O}\) of octonions. This density is characterised by the Vandermonde determinant structure and the exponential weight that is dependent on the trace of the given matrix.

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Acknowledgements

The financial support for this research by the Swedish International Development Agency, (Sida), Grant No.316, International Science Program, (ISP) in Mathematical Sciences, (IPMS) is gratefully acknowledged. Asaph Muhumuza is also grateful to the research environment Mathematics and Applied Mathematics (MAM), Division of Mathematics and Physics, School of Education, Culture and Communication, Mälardalen University for providing an excellent and inspiring environment for research education and research. Sergei Silvestrov is grateful to the Royal Swedish Academy of Sciences for partial support.

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Authors and Affiliations

  1. Department of Mathematics, Busitema University, Box 236, Tororo, Uganda

    Asaph Keikara Muhumuza

  2. Division of Mathematics and Physics, School of Education, Culture and Communication, Mälardalen University, Box 883, 72123, Västerås, Sweden

    Karl Lundengård, Anatoliy Malyarenko & Sergei Silvestrov

  3. Department of Mathematics, Makerere University, Box 7062, Kampala, Uganda

    John Magero Mango & Godwin Kakuba

Authors
  1. Asaph Keikara Muhumuza
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  2. Karl Lundengård
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  3. Anatoliy Malyarenko
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  4. Sergei Silvestrov
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  5. John Magero Mango
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  6. Godwin Kakuba
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Corresponding author

Correspondence to Sergei Silvestrov .

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Editors and Affiliations

  1. Division of Mathematics and Physics, Mälardalen University, Västerås, Sweden

    Sergei Silvestrov

  2. Division of Mathematics and Physics, Mälardalen University, Västerås, Sweden

    Anatoliy Malyarenko

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Muhumuza, A.K., Lundengård, K., Malyarenko, A., Silvestrov, S., Mango, J.M., Kakuba, G. (2023). The Wishart Distribution on Symmetric Cones. In: Silvestrov, S., Malyarenko, A. (eds) Non-commutative and Non-associative Algebra and Analysis Structures. SPAS 2019. Springer Proceedings in Mathematics & Statistics, vol 426. Springer, Cham. https://doi.org/10.1007/978-3-031-32009-5_23

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