Poisson limits for a hard-core clustering model

Let X1n,...,X>nn denote the locations of n points in a bounded, [gamma]-dimensional, Euclidean region Dn which has positive [gamma]-dimensional Lebesgue measure [mu](Dn). Let {Yn(r): r> 0} be the interpoint distance process for these points where Yn(r) is the number of pairs of points(Xin, Xin) which with i < j have Euclidean distance ||Xin - X>in|| < r. In this article we study the limiting distribution of Yn(r) when n --> [infinity] and [mu](Dn) --> [infinity], and the joint density of X1n,...,Xnnis of the form where r0 is a positive constant and Cn 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 ||Xin - Xin|| < r0) 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 < r0 < r < r00 provided sparseness conditions discussed by Saunders and Funk [9] hold. The sparseness conditions which require [mu](Dn)/n2 converges to a positive constant and the boundary of Dn 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 [less-than-or-equals, slant] 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.