Extreme Value Theory for Random Exponentials

We study the limit distribution of upper extreme values of i.i.d. exponential samples {ei}i=1 as t →∞, N →∞. Two cases are considered: (A) ess sup X = 0 and (B) ess sup X = ∞. We assume that the function h(x) = − log P{X > x} (case B) or h(x) = − log P{X > −1/x} (case A) is (normalized) regularly varying at ∞ with index 1 < % < ∞ (case B) or 0 < % < ∞ (case A). The growth scale of N is chosen in the form N ∼ eλH0(t) (0 < λ < ∞), where H0(t) is a certain asymptotic version of the function H(t) := log E[etX ] (case B) or H(t) = − log E[etX ] (case A). As shown earlier by Ben Arous et al. [5], there are critical points λ1 < λ2, below which the LLN and CLT, respectively, break down, whereas for 0 < λ < λ2 the limit laws for the sum SN (t) = e tX1 + · · · + etXN prove to be stable, with characteristic exponent α = α(%, λ) ∈ (0, 2). In this paper, we obtain the (joint) limit distribution of the upper order statistics of the exponential sample. In particular, M1,N = max{ei}i=1 has asymptotically the Fréchet distribution with parameter α. We also show that the empirical extremal measure converges (in fdd) to a Poisson random measure with intensity d(x−α). These results are complemented by explicit representations of the joint limit distribution of SN (t) and M1,N (t) (and in particular of their ratio) in terms of i.i.d. random variables with standard exponential distribution.