1. Let η ,...,η be n positive random variables (we do not assume their independence, in general). Prove that the distribution of η + ... + η is heavy-tailed if 1 n 1 n and only if the distribution of at least one of the summands is heavy-tailed. 2. Suppose that ξ , . . . , 1 ξ are n independent random variables with a common distribution F and that ξ(1) ≤ ξ(2) ≤ ... ≤ ξ(n) are the order statistics. (i) For k ≤ n, prove that the distribution of ξ(k) is heavy-tailed if and only if F is heavy-tailed. (ii) For k ≤ n − 1, prove that the distribution of ξ(k+1) − ξ(k) is heavy-tailed if and only if F is heavy-tailed. (iii) Based on (ii) and on Problem 1, prove that ξ(k) − ξ(l) has a heavy-tailed distribution if and only if F is heavy-tailed. Lemma 2.26. Suppose that the distribution F is long-tailed (F 2 L) and such that (2.21) holds. Then FI is long-tailed as well (FI 2 L) and F(x) = o(FI(x)) as x ! ¥. Proof. The long-tailedness of FI follows from the relations, as x ! ¥, FI(x+t) = Zx¥ F(x+t + y)dy ∼ Zx¥ F(x+ y)dy = FI(x); for any fixed t. That F(x) = o(FI(x)) as x ! ¥ now follows from Lemma 2.25. u t The converse assertion, that is, that long-tailedness of FI implies long-tailedness of F, is not in general true. This is illustrated by the following example. Please use Lemma 2.26 to solve previous problems