The structure of the class of subexponential distributions

LetX 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 byX 1:n ≦X 2:n ≦...≦X n:n the order statistics of the sample. We characterize the class of d.f.F for which $$P(X_{1:n} + X_{2:n} + \ldots + X_{n - i:n} > x) \sim P(X_{n - i:n} > x) as x \to \infty $$ for fixedn andi (i≦n-1), and we show that it is independent ofn. This leads to the genesis of a new class of d.f.L i ; we show that the sequence (L i ) ∞ =0 is strictly decreasing and we illustrate how the classesL i determine the probabilistic structure of the classL of subexponential distributions.