Compound Poisson approximation and the clustering of random points

Let n random points be uniformly and independently distributed in the unit square, and count the number W of subsets of k of the points which are cove red by some translate of a small square C. If n\C\ is small, the number of such clusters is approximately Poisson distributed, but the quality of the approximation is poor. In this paper, we show that the distribution of W ca n be much more closely approximated by an appropriate compound Poisson dist ribution CP(lambda(1), lambda(2),...). The argument is based on Stein's met hod, and is far from routine, largely because the approximating distributio n does not satisfy the simplifying condition that i lambda(i) be decreasing .