Probability Theory and Related Fields

, Volume 132, Issue 4, pp 579-612

First online:

Limit theorems for sums of random exponentials

  • Gérard Ben ArousAffiliated withCourant Institute of Mathematical Sciences Email author 
  • , Leonid V. BogachevAffiliated withDepartment of Statistics, University of Leeds
  • , Stanislav A. MolchanovAffiliated withDepartment of Mathematics, University of North Carolina at Charlotte

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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, 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.


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