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A new graph related to the directions of nearest neighbours in a point process

Part of: Limit theorems Geometric probability and stochastic geometry Stochastic processes Multivariate analysis

Published online by Cambridge University Press:  01 July 2016

S. N. Chiu  and
I. S. Molchanov
S. N. Chiu*
Affiliation:
Hong Kong Baptist University
I. S. Molchanov*
Affiliation:
University of Bern
*
∗ Postal address: Department of Mathematics, Hong Kong Baptist University, Kowloon, Hong Kong. Email address: snchiu@math.hkbu.edu.hk
∗∗ Postal address: Department of Mathematical Statistics and Actuarial Science, University of Bern, Sidlerstr. 5, CH-3012 Bern, Switzerland.
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Abstract

This paper introduces a new graph constructed from a point process. The idea is to connect a point with its nearest neighbour, then to the second nearest and continue this process until the point belongs to the interior of the convex hull of these nearest neighbours. The number of such neighbours is called the degree of a point. We derive the distribution of the degree of the typical point in a Poisson process, prove a central limit theorem for the sum of degrees, and propose an edge-corrected estimator of the distribution of the degree that is unbiased for a stationary Poisson process. Simulation studies show that this degree is a useful concept that allows the separation of clustering and repulsive behaviour of point processes.

Keywords

Point processrandom graphconvex hulldegree

MSC classification

Primary: 60G55: Point processes 60D05: Geometric probability and stochastic geometry
Secondary: 60F05: Central limit and other weak theorems 62H11: Directional data; spatial statistics
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 35 , Issue 1 , March 2003 , pp. 47 - 55
Copyright
Copyright © Applied Probability Trust 2003 

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References

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