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Limit theorems for sums of random exponentials
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  • Published: 10 February 2005

Limit theorems for sums of random exponentials

  • Gérard Ben Arous1,
  • Leonid V. Bogachev2 &
  • Stanislav A. Molchanov3 

Probability Theory and Related Fields volume 132, pages 579–612 (2005)Cite this article

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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 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.

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Author information

Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, 251 Mercer Street, New York NY, 10012, USA

    Gérard Ben Arous

  2. Department of Statistics, University of Leeds, Leeds, LS2 9JT, UK

    Leonid V. Bogachev

  3. Department of Mathematics, University of North Carolina at Charlotte, Charlotte, NC 28223, USA

    Stanislav A. Molchanov

Authors
  1. Gérard Ben Arous
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  2. Leonid V. Bogachev
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  3. Stanislav A. Molchanov
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Corresponding author

Correspondence to Gérard Ben Arous.

Additional information

Research supported in part by the DFG grants 436 RUS 113/534 and 436 RUS 113/722.

Mathematics Subject Classification (2000): 60G50, 60F05, 60E07

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Ben Arous, G., Bogachev, L. & Molchanov, S. Limit theorems for sums of random exponentials. Probab. Theory Relat. Fields 132, 579–612 (2005). https://doi.org/10.1007/s00440-004-0406-3

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  • Received: 19 February 2004

  • Revised: 23 October 2004

  • Published: 10 February 2005

  • Issue Date: July 2005

  • DOI: https://doi.org/10.1007/s00440-004-0406-3

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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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