Which negative multinomial distributions are infinitely divisible?

Abstract

We define a negative multinomial distribution on $\mathbb{N}^{n}_0$, where $\mathbb{N}_0$ is the set of non-negative integers, by its probability generating function which will be of the form $(A(a_{1},\rm dots,a_{n})/$ $A(a_{1}z_{2},\rm dots,a_{n}z_{n}))^{\lambdaambda}$, where \[A(\mathbf{z})=\sum\lambdaimits_{T\subset\lambdaeft\{1,2,\rm dots,n\right\}}a_{T}\prod\lambdaimits_{i\in T}z_{i},\] \(a_{\rm emptyset}\neq0\), and \(\lambdaambda\) is a positive number. Finding couples \(\lambdaeft( A,\lambdaambda\right)\) for which we obtain a probability generating function is a difficult problem. We establish necessary and sufficient conditions on the coefficients of \(A\) for which we obtain a probability generating function for any positive number \(\lambdaambda\). In consequence, we obtain all infinitely divisible negative multinomial distributions on $\mathbb{N}^{n}_0$.