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

Consider two independent homogeneous Poisson point processes Π of intensity λ and Π′ of intensity λ′ in d-dimensional Euclidean space. Let qk,d, k=0,1,…, be the fraction of Π-points which are the nearest Π-neighbor of precisely k   Π′-points. It is known that as d→∞, the qk,d converge to the Poisson probabilities e−λ′/λ(λ′/λ)k/k!, k=0,1,…. We derive the (sharp) rate of convergence d−1/2(4/33)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 Π. An extension to the case involving more than two independent Poisson point processes is also considered.