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April 2006 A characterization of Poisson-Gaussian families by generalized variance
Célestin C. Kokonendji, Afif Masmoudi
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Bernoulli 12(2): 371-379 (April 2006). DOI: 10.3150/bj/1145993979

Abstract

We show that if the generalized variance of an infinitely divisible natural exponential family F =F(μ) in a d -dimensional linear space is of the form det K μ ' '(θ )=exp(θ T b +c) , then there exists k in { 0,1,...,d} such that F is a product of k univariate Poisson and ( d -k )-variate Gaussian families. In proving this fact, we use a suitable representation of the generalized variance as a Laplace transform and the result, due to Jörgens, Calabi and Pogorelov, that any strictly convex smooth function f defined on the whole of R d such that det f ' '(θ ) is a positive constant must be a quadratic form.

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Célestin C. Kokonendji. Afif Masmoudi. "A characterization of Poisson-Gaussian families by generalized variance." Bernoulli 12 (2) 371 - 379, April 2006. https://doi.org/10.3150/bj/1145993979

Information

Published: April 2006
First available in Project Euclid: 25 April 2006

zbMATH: 1106.60014
MathSciNet: MR2218560
Digital Object Identifier: 10.3150/bj/1145993979

Rights: Copyright © 2006 Bernoulli Society for Mathematical Statistics and Probability

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Vol.12 • No. 2 • April 2006
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