Domain of existence of the Laplace transform of infinitely divisible negative multinomial distributions

Abstract

This article provides the domain of existence $\Omega$ of the Laplace transform of infinitely divisible negative multinomial distributions. We define a negative multinomial distribution on ${\mathbb{N}}^n,$ where $\mathbb{N}$ is the set of nonnegative integers, by its probability generating function which will be of the form ${\left(A\left(a_1{z}_1,\dots ,a_n{z}_n\right)/A\left(a_1,\dots ,a_n\right)\right)}^{-\lambda }$ where $A\left(\mathbf{z}\right)=\sum _{T\subset \left\{1,2,\dots ,n\right\}}{a}_T\prod _{i\in T}{z}_i,$ where $a_{\varnothing }\ne 0,$ and where $\lambda$ is a positive number. Finding couples $\left(A,\lambda \right)$ for which we obtain a probability generating function is a difficult problem. Necessary and sufficient conditions on the coefficients $a_T$ of $A$ for which we obtain a probability generating function for any positive number $\lambda$ are know by (Bernardoff, 2003). Thus we obtain necessary and sufficient conditions on $\mathbf{a}=\left(a_1,\dots ,a_n\right)$ so that $\mathbf{a}=\left({\mathbf{e}}^{t_1},\dots ,{\mathbf{e}}^{t_n}\right)$ with $\mathbf{t}=\left(t_1,\dots ,t_n\right)$ belonging to $\Omega$. This makes it possible to construct all the infinitely divisible multinomial distributions on ${\mathbb{N}}^n$. We give examples of construction in dimensions 2 and 3.