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Compound poisson approximations for the numbers of extreme spacings

Part of: Distribution theory Limit theorems Nonparametric inference Geometric probability and stochastic geometry Combinatorial probability

Published online by Cambridge University Press:  01 July 2016

Małgorzata Roos
Małgorzata Roos*
Affiliation:
University of Zurich
*
* Postal address: Institut für Angewandte Mathematik, Rämistr. 74, CH-8001 Zürich, Switzerland. E-mail address: k567833@czhrzula.bitnet
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Abstract

The accuracy of the Poisson approximation to the distribution of the numbers of large and small m-spacings, when n points are placed at random on the circle, was analysed using the Stein–Chen method in Barbour et al. (1992b). The Poisson approximation for m≧2 was found not to be as good as for 1-spacings. In this paper, rates of approximation of these distributions to suitable compound Poisson distributions are worked out, using the CP–Stein–Chen method and an appropriate coupling argument. The rates are better than for Poisson approximation for m≧2, and are of order O((log n)2/n) for large m-spacings and of order O(1/n) for small m-spacings, for any fixed m≧2, if the expected number of spacings is held constant as n → ∞.

Keywords

STEIN-CHEN METHODm-SPACINGSCOMPOUND POISSON DISTRIBUTIONRATE OF CONVERGENCECIRCULAR m-SCAN STATISTICSORDER STATISTICS

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60C05: Combinatorial probability 60D05: Geometric probability and stochastic geometry 62E20: Asymptotic distribution theory 62G30: Order statistics; empirical distribution functions
Type
Research Article
Information
Advances in Applied Probability , Volume 25 , Issue 4 , December 1993 , pp. 847 - 874
Copyright
Copyright © Applied Probability Trust 1993 

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Footnotes

This work was supported in part by the Schweizerischer Nationalfonds Grants Nos 21–25579.88 and 20–31262.91.

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