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Laplace transforms which are negative powers of quadratic polynomials. (English) Zbl 1152.60019
Summary: 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 S. T. Jensen [Ann. Stat. 16, No. 1, 302–322 (1988; Zbl 0653.62042)], and later on by J. Faraut and A. Korányi [Analysis on symmetric cones. Oxford: Clarendon Press (1994; Zbl 0841.43002)].

MSC:
60E10 Characteristic functions; other transforms
60E05 Probability distributions: general theory
44A10 Laplace transform
62E10 Characterization and structure theory of statistical distributions
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