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Compound Poisson approximation and the clustering of random points


Barbour, A D; Månsson, M (2000). Compound Poisson approximation and the clustering of random points. Advances in Applied Probability, 32(1):19-38.

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.

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.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords:Compound Poisson approximation; Stein's method
Language:English
Date:2000
Deposited On:04 Mar 2010 11:03
Last Modified:05 Apr 2016 13:25
Publisher:Applied Probability Trust
ISSN:0001-8678
Publisher DOI:10.1239/aap/1013540020
Related URLs:http://www.jstor.org/stable/1428224
Permanent URL: http://doi.org/10.5167/uzh-22064

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