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On improvements of the order of approximation in the Poisson limit theorem

Part of: Classical measure theory Groups and semigroups of linear operators, their generalizations and applications Limit theorems

Published online by Cambridge University Press:  14 July 2016

K. Borovkov  and
D. Pfeifer
K. Borovkov*
Affiliation:
University of Melbourne
D. Pfeifer*
Affiliation:
Universität Hamburg
*
∗Postal address: Department of Statistics, University of Melbourne, Parkville 3052, Australia.
∗∗Postal address: Universität Hamburg, Bundesstrasse 55, D-20146 Hamburg, Germany.
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Abstract

In this paper we consider improvements in the rate of approximation for the distribution of sums of independent Bernoulli random variables via convolutions of Poisson measures with signed measures of specific type. As a special case, the distribution of the number of records in an i.i.d. sequence of length n is investigated. For this particular example, it is shown that the usual rate of Poisson approximation of O(1/log n) can be lowered to O(1/n2). The general case is discussed in terms of operator semigroups.

Keywords

SIGNED MEASURESRECORDSOPERATOR SEMIGROUPS

MSC classification

Primary: 60F05: Central limit and other weak theorems 28A33: Spaces of measures, convergence of measures
Secondary: 47D03: Groups and semigroups of linear operators
Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 1 , March 1996 , pp. 146 - 155
Copyright
Copyright © Applied Probability Trust 1996 

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Footnotes

This research was partially supported by a grant of the Alexander von Humboldt Foundation, Germany.

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