Poisson approximation in connection with clustering of random points

Let n particles be independently and uniformly distributed in a rectangle A subset of R-2. Each subset consisting of k less than or equal to n particl es may possibly aggregate in such a way that it is covered by some translat e of a given convex set C subset of k The number of h-subsets which actuall y are covered by translates of C is denoted by W. The positions of such sub sets constitute a point process on k Each point of this process can be mark ed with the smallest necessary "size" of a set, of the same shape and orien tation as C, which covers the particles determining the point. This results in a marked paint process. The purpose of this paper is to consider Poisson (process) approximations o f W and of the above point processes, by means of Stein's method. To this e nd, the exact probability for It specific particles to be covered by some t ranslate of C is given.