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'Wald's Lemma' for Sums of Order Statistics of i.i.d. Random Variables

F. Thomas Bruss and James B. Robertson
Advances in Applied Probability
Vol. 23, No. 3 (Sep., 1991), pp. 612-623
DOI: 10.2307/1427625
Stable URL: http://www.jstor.org/stable/1427625
Page Count: 12
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'Wald's Lemma' for Sums of Order Statistics of i.i.d. Random Variables
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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 X(Nn) and X(n-Mn+1) as n→ ∞ , both for fixed s, and where s=sn is an increasing function of n.

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