Poisson limits for a hard-core clustering model

Roy Saunders, Richard J. Kryscio, Gerald M. Funk

Research output: Contribution to journalArticlepeer-review

13 Scopus citations


Let X1n,...,X>nn denote the locations of n points in a bounded, γ-dimensional, Euclidean region Dn which has positive γ-dimensional Lebesgue measure μ(Dn). Let {Yn(r): r > 0} be the interpoint distance process for these points where Yn(r) is the number of pairs of points(Xin, Xin) which with i < j have Euclidean distance {norm of matrix}Xin - X>in{norm of matrix} < r. In this article we study the limiting distribution of Yn(r) when n → ∞ and μ(Dn) → ∞, and the joint density of X1n,...,Xnnis of the form f{hook}(x1...x1)=Cn exp(vyn(r)) ifyn(r0)=0,0 ifyn(r0)>0 where r0 is a positive constant and Cn is a normalizing constant. These joint densities modify the Strauss [11] clustering model densities by introducing a hard-core component (no two points can have {norm of matrix}Xin - Xin{norm of matrix} < r0) found in the Matérn [4] models. In our main result we show that the interpoint distance process converges to a non-homogeneous Poisson process for r values in a bounded interval 0 < r0 < r < r00 provided sparseness conditions discussed by Saunders and Funk [9] hold. The sparseness conditions which require μ(Dn) n2 converges to a positive constant and the boundary of Dn is negligible are essentially equivalent to requiring that although the number of points n is large the region is large enough so that the points are sparse in this region. That is, it is rare for a point to have another point close to it. These results extend results for v ≤ 0 given by Saunders and Funk [9] where it is shown that without the hard core component such results do not hold for v > 0. Statistical applications are discussed.

Original languageEnglish
Pages (from-to)97-106
Number of pages10
JournalStochastic Processes and their Applications
Issue number1
StatePublished - Oct 1981


  • Clustering model
  • Poisson process
  • hard-core
  • raduis of influence
  • sparseness
  • weak convergence

ASJC Scopus subject areas

  • Statistics and Probability
  • Modeling and Simulation
  • Applied Mathematics


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