Limit Laws for Sums of Random Exponentials

We study the limiting distribution of the sum SN (t) = PN i=1 e tXi as t → ∞, N → ∞, where (Xi) 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) = − logP{Xi > x} is regularly varying at infinity with index 1 < ̺ < ∞. An appropriate scale for the growth of N relative to t is of the form eλH0(t), where the rate function H0(t) is a certain asymptotic version of the cumulant generating function H(t) = log E[ei ] 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(̺, λ) ranging from 0 to 2 and with skewness parameter β = 1. A limit theorem for the maximal value of the sample {ei , i = 1, . . . , N} is also proved.