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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Laplace transforms which are negative powers of quadratic polynomials


Authors: G. Letac and J. Wesołowski
Journal: Trans. Amer. Math. Soc. 360 (2008), 6475-6496
MSC (2000): Primary 60E05, 44A10, 62E10
DOI: https://doi.org/10.1090/S0002-9947-08-04463-2
Published electronically: June 3, 2008
MathSciNet review: 2434295
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Abstract: We find the distributions in $\mathbb {R}^n$ for the independent random variables $X$ and $Y$ such that $\mathbb {E}(X|X+Y)=a(X+Y)$ and $\mathbb {E}(q(X)|X+Y)=bq(X+Y)$ where $q$ runs through the set of all quadratic forms on $\mathbb {R}^n$ orthogonal to a given quadratic form $v.$ The essential part of this class is provided by distributions with Laplace transforms $(1-2\langle c,s\rangle +v(s))^{-p}$ that we describe completely, obtaining a generalization of a Gindikin theorem. This leads to the classification of natural exponential families with the variance function of type $\frac {1}{p}m\otimes m-\varphi (m)M_v$, where $M_v$ is the symmetric matrix associated to the quadratic form $v$ and $m\mapsto \varphi (m)$ is a real function. These natural exponential families extend the classical Wishart distributions on Lorentz cones already considered by Jensen, and later on by Faraut and Korányi.


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Additional Information

G. Letac
Affiliation: Laboratoire de Statistique et Probabilités, Université Paul Sabatier, 31062 Toulouse, France
Email: letac@cict.fr

J. Wesołowski
Affiliation: Wydział Matematyki i Nauk Informacyjnych, Politechnika Warszawska, Warszawa, Poland
Email: wesolo@mini.pw.edu.pl

Keywords: Characterizations of probabilities, Gindikin Theorem, Lorentz cone, Wishart distributions, natural exponential families, variance functions.
Received by editor(s): May 8, 2006
Received by editor(s) in revised form: December 1, 2006
Published electronically: June 3, 2008
Article copyright: © Copyright 2008 American Mathematical Society