Poisson approximation in connection with clustering of random points

Let n particles be independently and uniformly distributed in a rectangle A c R2. Each subset consisting of k < n particles may possibly aggregate in such a way that it is covered by some translate of a given convex set C c A. The number of k-subsets which actually are covered by translates of C is denoted by W. The positions of such subsets constitute a point process on A. Each point of this process can be marked with the smallest necessary "size" of a set, of the same shape and orientation as C, which covers the particles determining the point. This results in a marked point process. The purpose of this paper is to consider Poisson (process) approximations of W and of the above point processes, by means of Stein's method. To this end, the exact probability for k specific particles to be covered by some translate of C is given. 1. Introduction. Assume n particles are uniformly and independently distributed in a rectangle A c R 2, and let C c A be a convex set, small relative to A. To avoid problems with the boundaries of A, the torus convention will be used throughout the paper. The aim of this paper is to investigate Poisson approximation of certain variables and point processes which concern