A characterization of Poisson-Gaussian families by generalized variance

Abstract

We show that if the generalized variance of an infinitely divisible natural exponential family $F=F\left(\mu \right)$ in a $d$ -dimensional linear space is of the form $det{K}_{\mu }^{\text{'}\text{'}}\left(\theta \right)=exp({\theta }^{T}b+c)$, 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 $det{f}^{\text{'}\text{'}}\left(\theta \right)$ is a positive constant must be a quadratic form.