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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
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.
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- Copyright © Applied Probability Trust 1986
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