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

This article provides the domain of existence $\mathrm{\Omega }$ 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,$andwhere$\lambda$isapositivenumber\mathrm{.}Findingcouples$\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 $$\mathbf{a}$=\left( a_{1},\ldots,a_{n}\right) $sothat$$\mathbf{a}$=\left( $\mathbf{e}$^{t_{1}},\ldots,$\mathbf{e}$^{t_{n}}\right) $with$$\mathbf{t}$=\left( t_{1},\ldots,t_{n}\right) $belongingto$\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.