On the Count Probability of Many Correlated Symmetric Events

TL;DR

This paper investigates the probability distribution of the number of correlated symmetric events occurring, deriving the characteristic function of the limiting distribution when events are correlated up to a finite order.

Contribution

It extends the understanding of count probabilities by deriving the characteristic function for correlated symmetric events with finite-order correlations.

Findings

Limiting count distribution characterized by a specific characteristic function.

Poisson distribution emerges for independent events as N approaches infinity.

Correlation up to finite order affects the count probability distribution.

Abstract

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.