Which multivariate gamma distributions are infinitely divisible?

Abstract

We define a multivariate gamma distribution on $R^n$ by its Laplace transform ${\left(P(-\theta )\right)}^{-\lambda }$, $\lambda >0,$ where

$P\left(\theta \right)={\sum }_{T\subset \{1,\dots ,n\}}{p}_T{\prod }_{i\in T}{\theta }_i.$

Under $p_{\{1,\dots ,n\}}\ne 0$, we establish necessary and sufficient conditions on the coefficients of $P,$ such that the above function is the Laplace transform of some probability distribution, for all $\lambda >0,$ thus characterizing all infinitely divisible multivariate gamma distributions on $R^n.$