, Volume 132, Issue 4, pp 579-612
Date: 10 Feb 2005

Limit theorems for sums of random exponentials

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Abstract.

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 H 0(t) is a certain asymptotic version of the function (case B) or (case A). We have found two critical points, λ12, 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