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A characterization of Wishart distribution and its generalization on symmetric cones through independence of some statistics.

Le : 06/07/2015 11h00
Par : Bartosz Kolodziejek (E.Polytecnique de Varsovie)
Lieu : I 103
Lien web :
Résumé : In the talk I will give new results regarding characterization theorems on Wishart distribution on symmetric cones. We will consider independent random variables $X$ and $Y$ valued in an irreducible symmetric cone $\Omega$ with some additional technical assumptions on the distribution of $X$ and $Y$. We will show that if the quotient $U=g(X+Y)X$ (function $g\colon\Omega\to G$, which satisfies $g(x)x=I$ is called a division algorithm) and $V=X+Y$ are independent, then $X$ and $Y$ have generalized Wishart distributions with suitable parameters. This is a generalization of Lukacs-Olkin-Rubin theorem, where additional strong assumption of invariance under the group of automorphisms was imposed on the quotient $U$. If time permits, I will give also new results on similar characterization of Wishart distribution (in this case $U=X^{-1}-(X+Y)^{-1}$), which also has strong connection with the conditional independence structure of Wishart matrix. This characterization is related to the Matsumoto-Yor property of Wishart distribution.