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 $\left( A\left( a_{1}z_{1} ,\ldots,a_{n}z_{n}\right) /A\left( a_{1},\ldots,a_{n}\right) \right) ^{-\lambda}$ where $A\left( \mathbf{z}\right) = {\displaystyle\sum\limits_{T\subset\left\{ 1,2,\ldots,n\right\} }} a_{T}\prod\limits_{i\in T}z_{i},$ where $a_{\emptyset}\neq0,$ 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},\ldots,a_{n}\right) $ so that $\mathbf{a}=\left( \mathbf{e}^{t_{1}},\ldots,\mathbf{e}^{t_{n}}\right) $ with $\mathbf{t}=\left( t_{1},\ldots,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.