Generalized multivariate Hermite distributions and related point processes (Q688363)

The paper is concerned with the problem of characterizing functions of the form \[ G({\mathbf z})=\exp \left\{ \sum_{0 \leq{\mathbf k}'\mathbf{1} \leq m}a_{\mathbf k}({\mathbf z^ k}-1) \right\}, \] where \({\mathbf z}=[z_ 1,\dots,z_ n]'\), which are probability generating functions \(({\mathbf k} \in{\mathbf N}^ n,\;{\mathbf z}^{\mathbf k}=z_ 1^{k_ 1}z_ 2^{k_ 2}\dots z_ n^{k_ n})\). The corresponding distributions are called generalized multivariate Hermite distributions. They are concerned with conditions under which some of the coefficients \(a_{\mathbf k}\) may be negative, yet \(G({\mathbf z})\) remains a p.g. function. In this situation Cuppens' conditions are reduced to a more concrete form. There are discussed related results for point processes through some interesting examples.