Extreme value theory for random exponentials

We study the limit distribution of upper extreme values of i.i.d. exponential samples {e^(tX_i), i=1,...,N} as t->infty, N->infty. Two cases are considered: (A) ess supX = 0 and (B) ess supX = 1. 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 1 with index 1 < rho < infty (case B) or 0 < rho < 1 (case A). The growth scale of N is chosen in the form N = exp(lambda H_0^rho(t)) (0 < lambda < infty), where H_0(t) is a certain asymptotic version of the function H(t) := log E[e^(tX)] (case B) or H(t) = −log E[e^(tX)] (case A). As shown earlier by Ben Arous et al.(2005), there are critical points lambda_1 < lambda_2, below which the LLN and CLT, respectively, break down, whereas for 0 < lambda < 2 the limit laws for the sum S_N(t) = e^(tX_1) + ··· + e^(tX_N) prove to be stable, with characteristic exponent alpha = alpha(rho,lambda) in (0,2). In this paper, we obtain the (joint) limit distribution of the upper order statistics of the exponential sample. In particular, M_{1,N} = max{e^(tX_i), i=1,...,N} has asymptotically the Frechet distribution with parameter alpha. We also show that the empirical extremal measure converges (in fdd) to a Poisson random measure with intensity d(x^{−alpha}). These results are complemented by explicit representations of the joint limit distribution of S_N(t) and M_{1,N}(t) (and in particular of their ratio) in terms of i.i.d. random variables with standard exponential distribution.