A large-dimensional independent and identically distributed property for nearest neighbor counts in Poisson processes

Abstract

For an arbitrary point of a homogeneous Poisson point process in a d-dimensional Euclidean space, consider the number of Poisson points that have that given point as their rth nearest neighbor $(r = 1, 2, \dots)$. It is shown that as d tends to infinity, these nearest neighbor counts $(r = 1, 2, \dots)$ are iid asymptotically Poisson with mean 1. The proof relies on Rényi's characterization of Poisson processes and a representation in the limit of each nearest neighbor count as a sum of countably many dependent Bernoulli random variables.