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On the probability that a random point is the jth nearest neighbour to its own kth nearest neighbour

Published online by Cambridge University Press:  14 July 2016

Norbert Henze
Norbert Henze*
Affiliation:
University of Hannover
*
∗Postal address: Institut für Mathematische Stochastik, Universität Hannover, Welfengarten 1, D-3000 Hannover 1, W. Germany.
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Abstract

In a homogeneous Poisson process in Rd, consider an arbitrary point X and let Y be its kth nearest neighbour. Denote by Rk the rank of X in the proximity order defined by Y, i.e., Rk = j if X is the jth nearest neighbour to Y. A representation for Rk in terms of a sum of independent random variables is obtained, and the limiting distribution of Rk, as k →∞, is shown to be normal. This result generalizes to mixtures of Poisson processes.

Keywords

POISSON PROCESSMIXTURES OF POISSON PROCESSES
Type
Short Communications
Information
Journal of Applied Probability , Volume 23 , Issue 1 , March 1986 , pp. 221 - 226
Copyright
Copyright © Applied Probability Trust 1986 

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