Limit theorems for multivariate discrete distributions

According to the usual law of small numbers a multivariate Poisson distribution is derived by defining an appropriate model for multivariate Binomial distributions and examining their behaviour for large numbers of trials and small probabilities of marginal and simultaneous successes. The weak limit law is a generalization of Poisson's distribution to larger finite dimensions with arbitrary dependence structure. Compounding this multivariate Poisson distribution by a Gamma distribution results in a multivariate Pascal distribution which is again asymptotically multivariate Poisson. These Pascal distributions contain a class of multivariate geometric distributions. Finally the bivariate Binomial distribution is shown to be the limit law of appropriate bivariate hypergeometric distributions. Proving the limit theorems mentioned here as well as understanding the corresponding limit distributions becomes feasible by using probability generating functions.