On the Count Probability of Many Correlated Symmetric Events

We consider $N$ events that are defined on a common probability space. Those events shell have a common probability function that is symmetric with respect to interchanging the events. We ask for the probability distribution of the number of events that occur. If the probability of a single event is proportional to $1/N$ the resulting count probability is Poisson distributed in the limit of $N\rightarrow \infty$ for independent events. In this paper we calculate the characteristic function of the limiting count probability distribution for events that are correlated up to an arbitrary but finite order.