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‘Wald's Lemma' for sums of order statistics of i.i.d. random variables

Published online by Cambridge University Press:  01 July 2016

F. Thomas Bruss*
Affiliation:
University of California, Los Angeles
James B. Robertson*
Affiliation:
University of California, Santa Barbara
*
Present address: Departement Wiskunde, F 733, Vrije Universiteit Brussel, B-1050 Brussels, Belgium.
∗∗Postal address: Department of Statistics and Applied Probability, University of California, Santa Barbara, CA 93106, USA.

Abstract

Let X1, X2, · ··, Xn be positive i.i.d. random variables with known distribution function having a finite mean. For a given s ≥0 we define Nn = N(n, s) to be the largest number k such that the sum of the smallest k Xs does not exceed s, and Mn = M(n, s) to be the largest number k such that the sum of the largest k X's does not exceed s. This paper studies the precise and asymptotic behaviour of E(Nn), E(Mn), Nn, Mn, and the corresponding ‘stopped' order statistics and as n →∞, both for fixed s, and where s =sn is an increasing function of n.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 1991 

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Footnotes

Research supported by the grant ‘Automatic Decision Strategies' (No. ST2J-0227-C) of the European Community.

References

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