Compound Poisson approximation and the clustering of random points
A. D. Barbour, Marianne Månsson · Advances in Applied Probability · 2000
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 covered 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 can be much more closely approximated by an appropriate compound Poisson distribution CP(λ 1 , λ 2 ,…). The argument is based on Stein's method, and is far from routine, largely because the approximating distribution does not satisfy the simplifying condition that i λ i be decreasing.