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Spring 2018 Singular Riesz measures on symmetric cones
Abdelhamid Hassairi, Sallouha Lajmi
Adv. Oper. Theory 3(2): 337-350 (Spring 2018). DOI: 10.15352/AOT.1706-1183

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.

Citation

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Abdelhamid Hassairi. Sallouha Lajmi. "Singular Riesz measures on symmetric cones." Adv. Oper. Theory 3 (2) 337 - 350, Spring 2018. https://doi.org/10.15352/AOT.1706-1183

Information

Received: 21 June 2017; Accepted: 12 September 2017; Published: Spring 2018
First available in Project Euclid: 15 December 2017

zbMATH: 06848503
MathSciNet: MR3738215
Digital Object Identifier: 10.15352/AOT.1706-1183

Subjects:
Primary: ‎46G12
Secondary: 28A25

Rights: Copyright © 2018 Tusi Mathematical Research Group

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Vol.3 • No. 2 • Spring 2018
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