Article

Probability Theory and Related Fields

, Volume 132, Issue 4, pp 579-612

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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Abstract.

We study limiting distributions of exponential sums http://static-content.springer.com/image/art%3A10.1007%2Fs00440-004-0406-3/MediaObjects/s00440-004-0406-3flb1.gif 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 http://static-content.springer.com/image/art%3A10.1007%2Fs00440-004-0406-3/MediaObjects/s00440-004-0406-3flb2.gif , where the rate function H 0(t) is a certain asymptotic version of the function http://static-content.springer.com/image/art%3A10.1007%2Fs00440-004-0406-3/MediaObjects/s00440-004-0406-3flb3.gif (case B) or http://static-content.springer.com/image/art%3A10.1007%2Fs00440-004-0406-3/MediaObjects/s00440-004-0406-3flb4.gif (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