## The Annals of Applied Probability

### 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.

#### Article information

Source
Ann. Appl. Probab., Volume 6, Number 2 (1996), 561-571.

Dates
First available in Project Euclid: 18 October 2002

https://projecteuclid.org/euclid.aoap/1034968144

Digital Object Identifier
doi:10.1214/aoap/1034968144

Mathematical Reviews number (MathSciNet)
MR1398058

Zentralblatt MATH identifier
0870.60045

#### Citation

Yao, Yi-Ching; Simons, Gordon. A large-dimensional independent and identically distributed property for nearest neighbor counts in Poisson processes. Ann. Appl. Probab. 6 (1996), no. 2, 561--571. doi:10.1214/aoap/1034968144. https://projecteuclid.org/euclid.aoap/1034968144

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• CHAPEL HILL, NORTH CAROLINA 27599-3260
• TAIPEI, TAIWAN E-MAIL: simons@stat.unc.edu