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

• Published in: Statistics & probability letters Vol. 92; pp. 143 - 147
• Main Author: Yao, Yi-Ching
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.09.2014
• Subjects: Total variation distance; Typical cell; Typical point; Voronoi tessellation; secondary; Typical cell; Voronoi tessellation; Typical point; Total variation distance; primary
• ISSN: 0167-7152; 1879-2103
• DOI: 10.1016/j.spl.2014.05.014

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.