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Normal approximation for random sums

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  01 July 2016

A. D. Barbour  and
Aihua Xia
A. D. Barbour*
Affiliation:
Universität Zürich
Aihua Xia*
Affiliation:
The University of Melbourne
*
∗ Postal address: Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland. Email address: adb@amath.unizh.ch
∗∗ Postal address: Department of Mathematics and Statistics, The University of Melbourne, Parkville, VIC 3010, Australia. Email address: xia@ms.unimelb.edu.au
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Abstract

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In this paper, we adapt the very effective Berry-Esseen theorems of Chen and Shao (2004), which apply to sums of locally dependent random variables, for use with randomly indexed sums. Our particular interest is in random variables resulting from integrating a random field with respect to a point process. We illustrate the use of our theorems in three examples: in a rather general model of the insurance collective; in problems in geometrical probability involving stabilizing functionals; and in counting the maximal points in a two-dimensional region.

Keywords

Stein's methodBerry-Esseen boundpoint processrandom fieldlocal dependencetwo-dimensional maximum

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60G50: Sums of independent random variables; random walks 60G57: Random measures 60G60: Random fields

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 38 , Issue 3 , September 2006 , pp. 693 - 728
Copyright
Copyright © Applied Probability Trust 2006 

Footnotes

Partially supported by Schweizerischer Nationalfondsprojekt Nr. 20-67909.02.

Partially supported by the ARC Centre of Excellence for Mathematics and Statistics of Complex Systems.

References

Anscombe, F. J. (1952). Large-sample theory of sequential estimation. Proc. Camb. Phil. Soc. 48, 600–607.Google Scholar
Avram, F. and Bertsimas, D. (1993). On central limit theorems in geometrical probability. Ann. Appl. Prob. 3, 1033–1046.Google Scholar
Bai, Z.-D., Chao, C.-C., Hwang, H.-K. and Liang, W.-Q. (1998). On the variance of the number of maxima in random vectors and its applications. Ann. Appl. Prob. 8, 886–895.Google Scholar
Bai, Z.-D., Devroye, L., Hwang, H.-K. and Tsai, T.-H. (2005). Maxima in hypercubes. Random Structures Algorithms 27, 290–309.CrossRefGoogle Scholar
Bai, Z.-D., Hwang, H.-K., Liang, W.-Q. and Tsai, T.-H. (2001). Limit theorems for the number of maxima in random samples from planar regions. Electron. J. Prob. 6, 41 pp.Google Scholar
Baldi, P. and Rinott, Y. (1989). On normal approximations of distributions in terms of dependency graphs. Ann. Prob. 17, 1646–1650.CrossRefGoogle Scholar
Barbour, A. D. and Xia, A. (2001). The number of two-dimensional maxima. Adv. Appl. Prob. 33, 727–750.CrossRefGoogle Scholar
Barbour, A. D., Karoński, M. and Ruciński, A. (1989). A central limit theorem for decomposable random variables with applications to random graphs. J. Combinatorial Theory B 47, 125–145.Google Scholar
Baryshnikov, Yu. (2000). Supporting-points processes and some of their applications. Prob. Theory Relat. Fields 117, 163–182.Google Scholar
Bolthausen, E. (1984). An estimate of the remainder in the combinatorial central limit theorem. Z. Wahrscheinlichkeitsth. 66, 379–386.CrossRefGoogle Scholar
Chen, L. H. Y. and Shao, Q.-M. (2004). Normal approximation under local dependence. Ann. Prob. 32, 1985–1985.Google Scholar
Chen, L. H. Y. and Xia, A. (2004). Stein's method, Palm theory and Poisson process approximation. Ann. Prob. 32, 2545–2569.Google Scholar
Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
Dembo, A. and Rinott, Y. (1996). Some examples of normal approximations by Stein's method. In Random Discrete Structures (IMA Vol. Math. Appl. 76), eds Aldous, D. and Pemantle, R., sp, Berlin, pp. 25–44.Google Scholar
Devroye, L. (1993). Records, the maximal layer, and uniform distributions in monotone sets. Comput. Math. Appl. 25, 19–31.CrossRefGoogle Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events. Springer, Berlin.Google Scholar
Erickson, R. V. (1974). L 1 bounds for asymptotic normality of m-dependent sums using Stein's technique. Ann. Prob. 2, 522–529.CrossRefGoogle Scholar
Gnedenko, B. V. and Korolev, V. Yu. (1996). Random Summation. Limit Theorems and Applications. CRC Press, Boca Raton, FL.Google Scholar
Goovaerts, M. J. and Dhaene, J. (1996). The compound Poisson approximation for a portfolio of dependent risks. Insurance Math. Econom. 18, 81–85.Google Scholar
Kallenberg, O. (1983). Random Measures, 3rd edn. Academic Press, London.Google Scholar
Kläver, H. and Schmitz, N. (2006). An inequality for the asymmetry of distributions and a Berry–Esseen theorem for random summation. J. Inequal. Pure Appl. Math. 7, Article 2, 25 pp.Google Scholar
Matérn, B. (1986). Spatial Variation (Lecture Notes Statist. 36), 2nd edn. Springer, Berlin.CrossRefGoogle Scholar
Penrose, M. D. and Yukich, J. E. (2005). Normal approximation in geometric probability. In Stein's Method and Applications, eds Barbour, A. D. and Chen, L. H. Y., Singapore University Press/World Scientific, Singapore, pp. 37–58.Google Scholar
Rényi, A. (1960). On the central limit theorem for the sum of a random number of independent random variables. Acta Math. Acad. Sci. Hungar. 11, 97–102.Google Scholar
Rinott, Y. and Rotar, V. (1996). A multivariate CLT for local dependence with n −1/2log n rate, and applications to multivariate graph related statistics. J. Multivariate Anal. 56, 333–350.CrossRefGoogle Scholar
Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (1999). Stochastic Processes for Insurance and Finance. John Wiley, Chichester.CrossRefGoogle Scholar
Silvestrov, D. S. (2004). Limit Theorems for Randomly Stopped Stochastic Processes. sp, London.Google Scholar
Stein, C. (1972). A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In Proc. 6th Berkeley Symp. Math. Statist. Prob., Vol. 2, eds Le Cam, L. M., Neyman, J. and Scott, E. L., University of California Press, Berkeley, pp. 583–602.Google Scholar
Stein, C. (1986). Approximate Computation of Expectations (IMS Lecture Notes Monogr. Ser. 7). Institute of Mathematical Statistics, Hayward, CA.Google Scholar
Stoyan, D. and Stoyan, H. (1994). Fractals, Random Shapes and Point Fields. Methods of Geometrical Statistics. John Wiley, Chichester.Google Scholar