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Riesz Exponential Families on Symmetric Cones

Author

Listed:
  • A. Hassairi

    (Sfax University)

  • S. Lajmi

    (Sfax University)

Abstract

Let E be a simple Euclidean Jordan algebra of rank r and let Ω be its symmetric cone. Given a Jordan frame on E, the generalized power Δ s (−θ −1) defined on −Ω is the Laplace transform of some positive measure R s on E if and only if s is in a given subset Ξ of R r . The aim of this paper is to study the natural exponential families (NEFs) F(R s ) associated to the measures R s . We give a condition on s so that R s generates a NEF, we calculate the variance function of F(R s ) and we show that a NEF F on E is invariant by the triangular group if and only if there exists s in Ξ such that either F=F(R s ) or F is the image of F(R s ) under the map x↦−x.

Suggested Citation

  • A. Hassairi & S. Lajmi, 2001. "Riesz Exponential Families on Symmetric Cones," Journal of Theoretical Probability, Springer, vol. 14(4), pages 927-948, October.
    • Handle: RePEc:spr:jotpro:v:14:y:2001:i:4:d:10.1023_a:1012592618872
    DOI: 10.1023/A:1012592618872
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    Citations

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    Cited by:

    1. A. Hassairi & S. Lajmi & R. Zine, 2008. "A Characterization of the Riesz Probability Distribution," Journal of Theoretical Probability, Springer, vol. 21(4), pages 773-790, December.
    2. Bartosz Kołodziejek, 2016. "The Lukacs–Olkin–Rubin Theorem on Symmetric Cones Without Invariance of the “Quotient”," Journal of Theoretical Probability, Springer, vol. 29(2), pages 550-568, June.
    3. Gribisch, Bastian & Hartkopf, Jan Patrick, 2023. "Modeling realized covariance measures with heterogeneous liquidity: A generalized matrix-variate Wishart state-space model," Journal of Econometrics, Elsevier, vol. 235(1), pages 43-64.
    4. A. Hassairi & S. Lajmi, 2004. "Classification of Riesz Exponential Families on a Symmetric Cone by Invariance Properties," Journal of Theoretical Probability, Springer, vol. 17(3), pages 521-539, July.
    5. Abdelhamid Hassairi & Fatma Ktari & Raoudha Zine, 2022. "On the Gaussian representation of the Riesz probability distribution on symmetric matrices," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 106(4), pages 609-632, December.

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