Abstract
We show that if the generalized variance of an infinitely divisible natural exponential family in a -dimensional linear space is of the form , then there exists in such that is a product of univariate Poisson and ()-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 defined on the whole of such that is a positive constant must be a quadratic form.
Citation
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
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