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

TL;DR

This paper characterizes the domain where the Laplace transform exists for infinitely divisible negative multinomial distributions, providing conditions for their construction and examples in low dimensions.

Contribution

It determines the domain of the Laplace transform for these distributions and offers a method to construct all such infinitely divisible multinomial distributions.

Findings

Identifies the domain of the Laplace transform for the distributions.

Provides necessary and sufficient conditions for probability generating functions.

Includes explicit examples in 2 and 3 dimensions.

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.