A characterization of Poisson - Gaussian families by generalized variance

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 Kμ″(θ) = exp(θTb + 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 ℝd such that det f″(θ) is a positive constant must be a quadratic form. © 2006 ISI/BS.