Limit laws for sums of random exponentials

We study the limiting distribution of the sum S-N(t) = Sigma(i=1)(N) e(tXi) as t -> infinity, N -> infinity, where (X-i) are i.i.d. random variables. Attention to such exponential sums has been motivated by various problems in random media theory. Examples include the quenched mean population size of a colony of branching processes with random branching rates and the partition function of Derrida's Random Energy Model. In this paper, the problem is considered under the assumption that the log-tail distribution function h(x) = -log P{X-I > x} is regularly varying at infinity with index 1 < rho < infinity. An appropriate scale for the growth of N relative to t is of the form e(lambda H0(t)), where the rate function Ho(t) is a certain asymptotic version of the cumulant. generating function H(t) = log E[e(tXi)] provided by Kasahara's exponential Tauberian theorem. We have found two critical points, 0 < lambda(1) < lambda(2) < infinity, below which the Law of Large Numbers and the Central Limit Theorem, respectively, break down. Below lambda(2), we impose a slightly stronger condition of normalized regular variation of h. The limit laws here appear to be stable, with characteristic exponent alpha = alpha(rho, lambda) ranging from 0 to 2 and with skewness parameter beta = 1. A limit theorem for the maximal value of the sample {e(tXi), i = 1,...,N} is also proved.