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Compound Poisson approximation and the clustering of random points

Part of: Distribution theory Limit theorems Stochastic processes

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 theorems 60G55: Point processes
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.

References

[1] Aldous, D. (1989). Probability Approximations via the Poisson Clumping Heuristic. Springer, New York.CrossRefGoogle Scholar
[2] Alm, S. E. (1997). On the distribution of scan statistics of a two-dimensional Poisson process. Adv. Appl. Prob. 29, 1–18.CrossRefGoogle Scholar
[3] Barbour, A. D. (1997). Stein's method. In Encyclopaedia of Statistical Sciences, Update, Vol. 1, eds Kotz, S., Read, C. B. and Banks, D. L.. John Wiley, New York, pp. 513–521.Google Scholar
[4] Barbour, A. D. and Eagleson, G. K. (1984). Poisson convergence for dissociated statistics. J. Roy. Statist. Soc. Ser. B 46, 397–402.Google Scholar
[5] Barbour, A. D. and Utev, S. (1999). Compound Poisson approximation in total variation. Stoch. Proc. Appl. 82, 89–125.CrossRefGoogle Scholar
[6] Barbour, A. D., Chen, L. H. Y. and Loh, W.-L. (1992a). Compound Poisson approximation for non-negative random variables via Stein's method. Ann. Prob. 20, 1843–1866.CrossRefGoogle Scholar
[7] Barbour, A. D., Holst, L. and Janson, S. (1992b). Poisson Approximation. Oxford University Press.CrossRefGoogle Scholar
[8] Eggleton, P. and Kermack, W. O. (1944). A problem in the random distribution of particles. Proc. Roy. Soc. Edinb. Sect. A 62, 103–115.Google Scholar
[9] Janson, S. (1990). Poisson approximation for large deviations. Random Structures Algorithms 1, 221–230.CrossRefGoogle Scholar
[10] Loader, C. R. (1991). Large-deviation approximations to the distribution of scan statistics. Adv. Appl. Prob. 23, 751–771.CrossRefGoogle Scholar
[11] Mack, C. (1948). An exact formula for Q_k(n), the probable number of k-aggregates in a random distribution of n points. Phil. Mag. 39, 778–790.CrossRefGoogle Scholar
[12] Mack, C. (1949). The expected number of aggregates in a random distribution of n points. Proc. Cambridge Phil. Soc. 46, 285–292.CrossRefGoogle Scholar
[13] Månsson, M., (1999a). Poisson approximation in connection with clustering of random points. Ann. Appl. Prob. 9, 465–492.CrossRefGoogle Scholar
[14] Månsson, M., (1999b). On Poisson approximation for continuous multiple scan statistics in two dimensions. In Scan Statistics and Applications, eds Glaz, J. and Balakrishnan, N.. Birkhäuser, Boston, pp. 225–297.CrossRefGoogle Scholar
[15] Roos, M. (1993). Compound Poisson approximations for the numbers of extreme spacings. Adv. Appl. Prob. 25, 847–874.CrossRefGoogle Scholar
[16] Roos, M. (1994a). Stein's method for compound Poisson approximation: the local approach. Ann. Appl. Prob. 4, 1177–1187.CrossRefGoogle Scholar
[17] Roos, M. (1994b). Stein–Chen method for compound Poisson approximation: the coupling approach. In Probability Theory and Mathematical Statistics. Proc. Sixth Vilnius Conf., ed. Gigelionis, B. et al. TEV, Vilnius, pp. 645–660.Google Scholar
[18] Silberstein, L. (1945). The probable number of aggregates in random distributions of points. Phil. Mag. 36, 319–336.CrossRefGoogle Scholar
[19] Silverman, B. W. and Brown, T. C. (1978). Short distances, flat triangles and Poisson limits. J. Appl. Prob. 15, 815–825.CrossRefGoogle Scholar
[20] Silverman, B. W. and Brown, T. C. (1979). Rates of Poisson convergence for U-statistics. J. Appl. Prob. 16, 428–432.Google Scholar