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

Chiu, S. N.; Molchanov, Ilya (2003). A new graph related to the directions of nearest neighbours in a point process. Advances in applied probability, 35(1), pp. 47-55. Applied Probability Trust 10.1239/aap/1046366098

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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.

Item Type:

Journal Article (Original Article)


08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematical Statistics and Actuarial Science

UniBE Contributor:

Molchanov, Ilya


500 Science > 510 Mathematics




Applied Probability Trust




Ilya Molchanov

Date Deposited:

08 Aug 2016 16:20

Last Modified:

05 Dec 2022 14:57

Publisher DOI:


Additional Information:

In honor of Joseph Mecke



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