, · ··, 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
), 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.