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Laws of the iterated logarithm for sums of the middle portion of the sample

Published online by Cambridge University Press:  24 October 2008

Erich Haeusler  and
David M. Mason
Erich Haeusler
Affiliation:
University of Munich, Munich, West Germany
David M. Mason
Affiliation:
University of Munich, Munich, West Germany
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Abstract

Let X1, X2, …, be a sequence of independent random variables with common distribution function F in the domain of attraction of a stable law and, for each n ≥ 1, let X1, n ≤ … ≤ Xn, n denote the order statistics based on the first n of these random variables. It is shown that sums of the middle portion of the order statistics of the form , where (kn)n ≥ 1 is a sequence of non-negative integers such that kn → ∞ and kn/n → 0 as n → ∞ at an appropriate rate, can be normalized and centred so that the law of the iterated logarithm holds. The method of proof is based on the almost sure properties of weighted uniform empirical processes.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 101 , Issue 2 , March 1987 , pp. 301 - 312
Copyright
Copyright © Cambridge Philosophical Society 1987

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References

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