Limit theorems for sums of random exponentials

We study limiting distributions of exponential sums as t→∞, N→∞, where (X_i) are i.i.d. random variables. Two cases are considered: (A) ess sup X_i = 0 and (B) ess sup X_i = ∞. We assume that the function h(x)= -log P{X_i>x} (case B) or h(x) = -log P {X_i>-1/x} (case A) is regularly varying at ∞ with index 1 < ϱ <∞ (case B) or 0 < ϱ < ∞ (case A). The appropriate growth scale of N relative to t is of the form , where the rate function H₀(t) is a certain asymptotic version of the function (case B) or (case A). We have found two critical points, λ₁<λ₂, below which the Law of Large Numbers and the Central Limit Theorem, respectively, break down. For 0 < λ < λ₂, under the slightly stronger condition of normalized regular variation of h we prove that the limit laws are stable, with characteristic exponent α = α (ϱ, λ) ∈ (0,2) and skewness parameter β ≡ 1.Research supported in part by the DFG grants 436 RUS 113/534 and 436 RUS 113/722.Mathematics Subject Classification (2000): 60G50, 60F05, 60E07