Poisson approximation for point processes via monotone couplings (Q1814750)

The Stein-Chen bounds for the accuracy of Poisson approximation to the distribution of a sum \(W\) of dependent indicator random variables take a particularly simple form if the variables are either positively or negatively related [the reviewer, \textit{L. Holst} and \textit{S. Janson}, ``Poisson approximation'' (1992; Zbl 0746.60002), pp. 24-26]. The appropriate concepts when \(W\) is the total number of points in a point process are introduced, and analogous upper and lower bounds are established. The key idea is to compare the measure of the point process with its reduced Palm measures. A number of examples are given.