## Abstract

Let X_{1n},...,X_{>nn} denote the locations of n points in a bounded, γ-dimensional, Euclidean region D_{n} which has positive γ-dimensional Lebesgue measure μ(D_{n}). Let {Y_{n}(r): r > 0} be the interpoint distance process for these points where Y_{n}(r) is the number of pairs of points(X_{in}, X_{in}) which with i < j have Euclidean distance {norm of matrix}X_{in} - X_{>in}{norm of matrix} < r. In this article we study the limiting distribution of Y_{n}(r) when n → ∞ and μ(D_{n}) → ∞, and the joint density of X_{1n},...,X_{nn}is of the form f{hook}(x_{1}...x_{1})=^{Cn exp(vyn(r)) ifyn(r0)=0,}_{0 ifyn(r0)>0} where r_{0} is a positive constant and C_{n} 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}X_{in} - X_{in}{norm of matrix} < r_{0}) 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 < r_{0} < r < r_{00} provided sparseness conditions discussed by Saunders and Funk [9] hold. The sparseness conditions which require μ(D_{n}) n^{2} converges to a positive constant and the boundary of D_{n} 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 language | English |
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Pages (from-to) | 97-106 |

Number of pages | 10 |

Journal | Stochastic Processes and their Applications |

Volume | 12 |

Issue number | 1 |

DOIs | |

State | Published - Oct 1981 |

## Keywords

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

## ASJC Scopus subject areas

- Statistics and Probability
- Modeling and Simulation
- Applied Mathematics