Definition and characterization of multivariate negative binomial distribution

The probability generating function (pgf) of an n-variate negative binomial distribution is defined to be [[beta](s1,...,sn)]-k where [beta] is a polynomial of degree n being linear in each si and k > 0. This definition gives rise to two characterizations of negative binomial distributions. An n-variate linear exponential distribution with the probability function h(x1,...,xn)exp([Sigma]i=1n [theta]ixi)/f([theta]1,...,[theta]n) is negative binomial if and only if its univariate marginals are negative binomial. Let St, t = 1,..., m, be subsets of {s1,..., sn} with empty [intersection]t=1mSt. Then an n-variate pgf is of a negative binomial if and only if for all s in St being fixed the function is of the form of the pgf of a negative binomial in other s's and this is true for all t.