Advances in Operator Theory

Singular Riesz measures on symmetric cones

Abdelhamid Hassairi and Sallouha Lajmi

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

A fondamental theorem due to Gindikin [Russian Math. Surveys, 29 (1964), 1-89] says that the‎ ‎generalized power $\Delta_{s}(-\theta^{-1})$ defined on a symmetric‎ ‎cone is the Laplace transform of a positive measure $R_{s}$ if and ‎only if $s$ is in a given subset $\Xi$ of $\mathbb{R}^{r}$‎, ‎where $r$‎ ‎is the rank of the cone‎. ‎When $s$ is in a well defined part of‎ ‎$\Xi$‎, ‎the measure $R_{s}$ is absolutely continuous with respect to‎ ‎Lebesgue measure and has a known expression‎. ‎For the other elements‎ ‎$s$ of $\Xi$‎, ‎the measure $R_{s}$ is concentrated on the boundary of‎ ‎the cone and it has never been explicitly determined‎. ‎The aim of the‎ ‎present paper is to give an explicit description of the measure‎ ‎$R_{s}$ for all $s$ in $\Xi$‎. ‎The work is motivated by the‎ ‎importance of these measures in probability theory and in statistics‎ ‎since they represent a generalization of the class of measures‎ ‎generating the famous Wishart probability distributions‎.

Article information

Source
Adv. Oper. Theory, Volume 3, Number 2 (2018), 337-350.

Dates
Received: 21 June 2017
Accepted: 12 September 2017
First available in Project Euclid: 15 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.aot/1513328634

Digital Object Identifier
doi:10.15352/AOT.1706-1183

Mathematical Reviews number (MathSciNet)
MR3738215

Zentralblatt MATH identifier
06848503

Subjects
Primary: 46G12: Measures and integration on abstract linear spaces [See also 28C20, 46T12]
Secondary: 28A25: Integration with respect to measures and other set functions

Keywords
Jordan algebra symmetric cone generalized power Laplace transform Riesz measure

Citation

Hassairi, Abdelhamid; Lajmi, Sallouha. Singular Riesz measures on symmetric cones. Adv. Oper. Theory 3 (2018), no. 2, 337--350. doi:10.15352/AOT.1706-1183. https://projecteuclid.org/euclid.aot/1513328634


Export citation

References

  • J. Faraut and A. Korányi, Analysis on symmetric cones, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1994.
  • P. J. Forrester, N. C. Snaith, and J. J. M. Verbaarschot, Developments in random matrix theory, J. Phys. A 36 (2003), no. 12, R1–R10.
  • S. G. Gindikin, Analysis in homogeneous domains, (Russian) Uspehi Mat. Nauk 19 (1964), no. 4 (118) 3–92.
  • A. Hassairi and S. Lajmi, Riesz exponential families on symmetric cones, J. Theoret. Probab. 14 (2001), no. 4, 927–948.
  • A. Hassairi and S. Lajmi, Classification of Riesz exponential families on a symmetric cone by invariance properties, J. Theoret. Probab. 17 (2004), no. 3, 521–539.
  • M. Lassalle, Algèbre de Jordan et ensemble de Wallah(French) [Jordan algebras and Wallach sets], Invent. Math. 89 (1987), no. 2, 375–393.
  • H. Massam and E. Neher, On transformation and determinants of Wishart variables on symmetric cones, J. Theoret. Probab. 10 (1997), no. 4, 867–902.
  • M. S. Srivastava, Singular Wishart and multivariate beta distributions, Ann. Statist. 31 (2003), no. 5, 1537–1560.
  • H. Uhlig, On singular Wishart and singular multivariate beta distributions, Ann. Statist. 22 (1994), no. 1, 395–405.