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Normal approximation for random sums

Part of: Stochastic processes Limit theorems

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
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.

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