Limit Laws for Norms of IID Samples with Weibull Tails

We are concerned with the limit distribution of lt-norms RN (t) = ‖XN‖t (of order t) of samples XN = (X1, . . . , XN ) of i.i.d. positive random variables, as N → ∞, t → ∞. The problem was first considered by Schlather(10), but the case where {Xi} belong to the domain of attraction of Gumbel’s double exponential law (in the sense of extreme value theory) has largely remained open (even for an exponential distribution). In this paper, it is assumed that the log-tail distribution function h(x) = − log P{X1 > x} is regularly varying at infinity with index 0 < % <∞. We proceed from studying the limit distribution of the sums SN (t) = ∑N i=1 X t i , which is of interest in its own right. A proper growth scale of N relative to t appears to be of the form N ∼ eαt/% (0 < α < ∞). We show that there are two critical points, α1 = 1 and α2 = 2, below which the law of large numbers and the central limit theorem, respectively, break down. For α < 2, under a slightly stronger condition of normalized regular variation of h, we prove that the limit laws for SN (t) are stable, with characteristic exponent α ∈ (0, 2) and skewness parameter β ≡ 1. A complete picture of the limit laws for the norms RN (t) = SN (t)1/t is then derived. In particular, our results corroborate a conjecture in Ref. 10 regarding the “endpoints” α→∞, α→ 0.