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

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.