A Large-Dimensional IID Property for Nearest Neighbor Counts in Poisson Processes
نویسندگان
چکیده
For an arbitrary point of a homogeneous Poisson point process in a d-dimen-siona! Euclidean space, consider the number of Poisson points that have that given point as their r-th nearest neighbor (r = 1,2,...). It is shown that as d tends to infinity, these nearest neighbor counts (r = 1,2,...) are tid asymptotically Poisson with mean 1. The proof relies on Renyi'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.
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