Laplace transforms which are negative powers of quadratic polynomials

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.