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Compound Poisson approximation and the clustering of random points

Part of:Stochastic processesLimit theoremsDistribution theory

Published online by Cambridge University Press:  01 July 2016

A. D. Barbour and
Marianne Månsson
A. D. Barbour*
Affiliation:
Universität Zürich
Marianne Månsson*
Affiliation:
Chalmers University of Technology
*
∗ 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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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 iλi be decreasing.

Keywords

Compound Poisson approximationStein's method

MSC classification

Primary:62E17: Approximations to distributions (nonasymptotic)60F05: Central limit and other weak theorems60G55: Point processes

Information

Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 32 , Issue 1 , March 2000, pp. 19 - 38
Copyright
Copyright © Applied Probability Trust 2000 

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Footnotes

Partially supported by Schweizerischer Nationalfonds Grant Nr. 20 - 50686.97.

Partially supported by the Swedish Natural Science Research Council.

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