Probability Theory and Related Fields

, Volume 132, Issue 4, pp 579–612

# Limit theorems for sums of random exponentials

• Gérard Ben Arous
• Leonid V. Bogachev
• Stanislav A. Molchanov
Article

## Abstract.

We study limiting distributions of exponential sums Open image in new window 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 Open image in new window , where the rate function H0(t) is a certain asymptotic version of the function Open image in new window (case B) or Open image in new window (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.

### Keywords

Sums of independent random variables Random exponentials Regular variation Exponential Tauberian theorems Central limit theorem Weak limit theorems Infinitely divisible distributions Stable laws

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## Authors and Affiliations

• Gérard Ben Arous
• 1
• Leonid V. Bogachev
• 2
• Stanislav A. Molchanov
• 3
1. 1.Courant Institute of Mathematical SciencesNew York NYUSA
2. 2.Department of StatisticsUniversity of LeedsLeedsUK
3. 3.Department of MathematicsUniversity of North Carolina at CharlotteCharlotteUSA