Poisson limits for a hard-core clustering model

Abstract

Let X 1 n ,…, X > nn denote the locations of n points in a bounded, γ-dimensional, Euclidean region D n which has positive γ-dimensional Lebesgue measure μ( D n ). Let { Y n ( r): r > 0} be the interpoint distance process for these points where Y n ( r) is the number of pairs of points( X in , X in ) which with i < j have Euclidean distance ‖ X in − X > in ‖ < r. In this article we study the limiting distribution of Y n ( r) when n → ∞ and μ( D n ) → ∞, and the joint density of X 1 n ,…, X nn is of the form ƒ(x 1…x 1)= C n exp(vy n(r)) ify n(r 0)=0, 0 ify n(r 0)>0 where r 0 is a positive constant and C n is a normalizing constant. These joint densities modify the Strauss [11] clustering model densities by introducing a hard-core component (no two points can have ‖ X in − X in ‖ < r 0) found in the Matérn [4] models. In our main result we show that the interpoint distance process converges to a non-homogeneous Poisson process for r values in a bounded interval 0 < r 0 < r < r 00 provided sparseness conditions discussed by Saunders and Funk [9] hold. The sparseness conditions which require μ(D n) n 2 converges to a positive constant and the boundary of D n is negligible are essentially equivalent to requiring that although the number of points n is large the region is large enough so that the points are sparse in this region. That is, it is rare for a point to have another point close to it. These results extend results for v ⩽ 0 given by Saunders and Funk [9] where it is shown that without the hard core component such results do not hold for v > 0. Statistical applications are discussed.