Rate of Poisson approximation for nearest neighbor counts in large-dimensional Poisson point processes

Consider two independent homogeneous Poisson point processes Pi of intensity lambda and Pi ' of intensity lambda ' in d-dimensional Euclidean space. Let q(k,d), k = 0, 1,..., be the fraction of Pi-points which are the nearest Pi-neighbor of precisely k Pi '-points. It is known that as d -> infinity, the q(k,d) converge to the Poisson probabilities e(-lambda '/lambda) (lambda '/lambda)(k)/k! k = 0, 1,.... We derive the (sharp) rate of convergence d(-1/2) (4/3 root 3)(d) which is related to the asymptotic behavior of the variance of the volume of the typical cell of the Poisson-Voronoi tessellation generated by Pi. An extension to the case involving more than two independent Poisson point processes is also considered. (C) 2014 Elsevier B.V. All rights reserved.