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May 1996 A large-dimensional independent and identically distributed property for nearest neighbor counts in Poisson processes
Yi-Ching Yao, Gordon Simons
Ann. Appl. Probab. 6(2): 561-571 (May 1996). DOI: 10.1214/aoap/1034968144

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.

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Yi-Ching Yao. Gordon Simons. "A large-dimensional independent and identically distributed property for nearest neighbor counts in Poisson processes." Ann. Appl. Probab. 6 (2) 561 - 571, May 1996. https://doi.org/10.1214/aoap/1034968144

Information

Published: May 1996
First available in Project Euclid: 18 October 2002

zbMATH: 0870.60045
MathSciNet: MR1398058
Digital Object Identifier: 10.1214/aoap/1034968144

Subjects:
Primary: 60G55
Secondary: 60B12 , 60D05

Keywords: Nearest neighbor counts , Poisson point process

Rights: Copyright © 1996 Institute of Mathematical Statistics

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Vol.6 • No. 2 • May 1996
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