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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Laplace transforms which are negative powers of quadratic polynomials
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by G. Letac and J. Wesołowski PDF
Trans. Amer. Math. Soc. 360 (2008), 6475-6496 Request permission

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.
References
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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
  • Received by editor(s): May 8, 2006
  • Received by editor(s) in revised form: December 1, 2006
  • Published electronically: June 3, 2008
  • © Copyright 2008 American Mathematical Society
  • 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
  • MathSciNet review: 2434295