Hostname: page-component-77c89778f8-vpsfw Total loading time: 0 Render date: 2024-07-22T23:30:51.006Z Has data issue: false hasContentIssue false

‘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*
University of California, Los Angeles
James B. Robertson*
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.


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.

Research Article
Copyright © Applied Probability Trust 1991 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


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


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, 454473.CrossRefGoogle Scholar
Samuels, S. M. and Steele, J. M. (1981) Optimal selection of a monotone sequence from a random sample. Ann. Prob. 9, 937947.Google Scholar
Wald, A. (1947) Sequential Analysis. Dover, New York.Google Scholar