Singular Riesz measures on symmetric cones

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
Accepted: 12 September 2017
First available in Project Euclid: 15 December 2017

https://projecteuclid.org/euclid.aot/1513328634

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

Mathematical Reviews number (MathSciNet)
MR3738215

Zentralblatt MATH identifier
06848503

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

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