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

Abstract

Let X 1 , X 2 , · ··, X n be positive i.i.d. random variables with known distribution function having a finite mean. For a given s ≥0 we define N n = N ( n, s ) to be the largest number k such that the sum of the smallest k Xs does not exceed s , and M n = 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 ( N n ), E ( M n ) , N n , M n , and the corresponding ‘stopped' order statistics and as n →∞, both for fixed s , and where s =s n is an increasing function of n.