## 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