On interpoint distances for planar Poisson cluster processes

For stationary Poisson or Poisson cluster processes ? on R2 we study the distribution of the interpoint distances using the interpoint distance function Y, (r) = f (S, (x) n D)d?(x) and the nearest-neighbor indicator function WD,(t) = f1,,)x{(S,(x)n D) = l}dJ(x). Here S, (x) is the interior of a circle of radius r having center x, I(t) is that subset of D which has x E D and S, (x) C D and X is the usual indicator function. We show that if the region D C R 2 is large, then these functions are approximately distributed as Poisson processes indexed by r -p/Vpl(D) and t -cV'log(1i(D)/r), where li(D) is the Lebesgue measure of D. POINT PROCESS; NEAREST-NEIGHBOUR DISTANCE; WEAK CONVERGENCE; TESTS FOR SPATIAL RANDOMNESS