Hassairi, A., Lajmi, S. (2018). Singular Riesz measures on symmetric cones. Advances in Operator Theory, 3(2), 337-350. doi: 10.15352/aot.1706-1183
Abdelhamid Hassairi; Sallouha Lajmi. "Singular Riesz measures on symmetric cones". Advances in Operator Theory, 3, 2, 2018, 337-350. doi: 10.15352/aot.1706-1183
Hassairi, A., Lajmi, S. (2018). 'Singular Riesz measures on symmetric cones', Advances in Operator Theory, 3(2), pp. 337-350. doi: 10.15352/aot.1706-1183
Hassairi, A., Lajmi, S. Singular Riesz measures on symmetric cones. Advances in Operator Theory, 2018; 3(2): 337-350. doi: 10.15352/aot.1706-1183
Receive Date: 21 June 2017,
Revise Date: 08 September 2017,
Accept Date: 12 September 2017
Abstract
A fondamental theorem due to Gindikin 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 $\Bbb{R}^{r}$, 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.
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