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

S. N. Chiu*, I. S. Molchanov

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

10 Citations (Scopus)
20 Downloads (Pure)


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.

Original languageEnglish
Pages (from-to)47-55
Number of pages9
JournalAdvances in Applied Probability
Issue number1
Publication statusPublished - Mar 2003

Scopus Subject Areas

  • Statistics and Probability
  • Applied Mathematics

User-Defined Keywords

  • Point process
  • random graph
  • convex hull
  • degree


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