Limit laws for sums of random exponentials

We study the limiting distribution of the sum as t → ∞, N → ∞, 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 < ϱ < ∞. An appropriate scale for the growth of N relative to t is of the form e^λH₀(t), where the rate function H₀(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< λ₁< λ₂ < ∞, below which the Law of Large Numbers and the Central Limit Theorem, respectively, break down. Below λ₂, we impose a slightly stronger condition of normalized regular variation of h. The limit laws here appear to be stable, with characteristic exponent α = α(ϱ, λ) ranging from 0 to 2 and with skewness parameter β = 1. A limit theorem for the maximal value of the sample {e^tXi, i = 1, ..., N} is also proved.