Let \(X_ 1,X_ 2,...,X_ n\) be a sequence of positive, independent, identically distributed random variables with the same distribution function (d.f.) F and denote by \(X_{1:n}\leq X_{2:n}\leq...\leq X_{n:n}\) the order statistics of the sample. We characterize the class of d.f. F for which \[ P(x_{1:n}+X_{2:n}+...+X_{n-i:n}>x)\sim P(x_{n-i:n}>x)\quad as\quad x\to \infty \] for fixed n and i (i\(\leq n- 1)\), and we show that it is independent of n. This leads to the genesis of a new class of d.f. \({\mathcal S}_ i\); we show that the sequence (\({\mathcal S}_ i)^{\infty}_{i=0}\) is strictly decreasing and we illustrate how the classes \({\mathcal S}_ i\) determine the probabilistic structure of the class \({\mathcal S}\) of subexponential distributions.