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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  and
James B. Robertson
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.
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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.

Keywords

STOPPING TIMESEMPIRICAL DISTRIBUTION FUNCTIONPROPHET INEQUALITY
Type
Research Article
Information
Advances in Applied Probability , Volume 23 , Issue 3 , September 1991 , pp. 612 - 623
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

Bruss, F. T. (1983) Resource dependent branching processes (abstract). Stoch. Proc. Appl. 16, 36.Google Scholar
Coffman, E. G., Flatto, L. and Weber, R. R. (1987) Optimal selection of stochastic integrals under a sum constraint. Adv. Appl. Prob. 19, 454–473.CrossRefGoogle Scholar
Samuels, S. M. and Steele, J. M. (1981) Optimal selection of a monotone sequence from a random sample. Ann. Prob. 9, 937–947.Google Scholar
Wald, A. (1947) Sequential Analysis. Dover, New York.Google Scholar