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Compound Poisson approximation and the clustering of random points

  • A. D. Barbour (a1) and Marianne Månsson (a2)
Abstract

Let n random points be uniformly and independently distributed in the unit square, and count the number W of subsets of k of the points which are covered by some translate of a small square C. If n|C| is small, the number of such clusters is approximately Poisson distributed, but the quality of the approximation is poor. In this paper, we show that the distribution of W can be much more closely approximated by an appropriate compound Poisson distribution CP(λ1, λ2,…). The argument is based on Stein's method, and is far from routine, largely because the approximating distribution does not satisfy the simplifying condition that iλ i be decreasing.

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Postal address: Institut für Angewandte Mathematik, Universität Zürich, Winterthurerstr. 190, CH-8057 Zürich, Switzerland.
∗∗ Postal address: Department of Mathematics, Chalmers University of Technology, SE-41296 Göteborg, Sweden.
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Partially supported by Schweizerischer Nationalfonds Grant Nr. 20 - 50686.97.

Partially supported by the Swedish Natural Science Research Council.

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Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
  • URL: /core/journals/advances-in-applied-probability
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