Domain of existence of the Laplace transform of negative multinomial distributions and simulations
A probability distribution \( \sum_{\boldsymbol{\alpha} \in \mathbb{N}^n} p_{\boldsymbol{\alpha}}\delta_{\boldsymbol{\alpha}}\) on the set of nonnegative integers \(\mathbb{N}^n\) is said to be a negative multinomial distribution if there exists an affine polynomial \(P(z_1,\dots , z_n)\) and \(\lambda >0\) such that \[ \sum_{\boldsymbol{\alpha} \in \mathbb{N}^n} p_{\boldsymbol{\alpha}} z_1^{\alpha_1}\, \dots z_n^{\alpha_n} =\left(P(z_1,\dots , z_n)\right)^{-\lambda}. \] In [Bernoulli 9, No. 5, 877--893 (2003; Zbl 1065.60012)] the same author characterized all affine polynomials \(P\) which give rise to probability generating functions for infinitely divisible negative multinomial distributions. In this paper the author uses this characterization to derive the domain of existence of their Laplace transforms. As an application of these results, the author provides a construction and a simulation of these distributions in dimension 2.
scientific article; zbMATH DE number 7629091