Bernoulli

  • Bernoulli
  • Volume 12, Number 2 (2006), 371-379.

A characterization of Poisson-Gaussian families by generalized variance

Célestin C. Kokonendji and Afif Masmoudi

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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.

Article information

Source
Bernoulli, Volume 12, Number 2 (2006), 371-379.

Dates
First available in Project Euclid: 25 April 2006

Permanent link to this document
https://projecteuclid.org/euclid.bj/1145993979

Digital Object Identifier
doi:10.3150/bj/1145993979

Mathematical Reviews number (MathSciNet)
MR2218560

Zentralblatt MATH identifier
1106.60014

Keywords
affine variance function determinant infinitely divisible measure Laplace transform Monge-Ampère equation r-reducibility

Citation

Kokonendji, Célestin C.; Masmoudi, Afif. A characterization of Poisson-Gaussian families by generalized variance. Bernoulli 12 (2006), no. 2, 371--379. doi:10.3150/bj/1145993979. https://projecteuclid.org/euclid.bj/1145993979


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