Probability Theory and Related Fields

, Volume 132, Issue 4, pp 579–612

Limit theorems for sums of random exponentials

Authors

    • Courant Institute of Mathematical Sciences
  • Leonid V. Bogachev
    • Department of StatisticsUniversity of Leeds
  • Stanislav A. Molchanov
    • Department of MathematicsUniversity of North Carolina at Charlotte
Article

DOI: 10.1007/s00440-004-0406-3

Cite this article as:
Ben Arous, G., Bogachev, L. & Molchanov, S. Probab. Theory Relat. Fields (2005) 132: 579. doi:10.1007/s00440-004-0406-3

Abstract.

We study limiting distributions of exponential sums https://static-content.springer.com/image/art%3A10.1007%2Fs00440-004-0406-3/MediaObjects/s00440-004-0406-3flb1.gif as t→∞, N→∞, where (Xi) are i.i.d. random variables. Two cases are considered: (A) ess sup Xi = 0 and (B) ess sup Xi = ∞. We assume that the function h(x)= -log P{Xi>x} (case B) or h(x) = -log P {Xi>-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 https://static-content.springer.com/image/art%3A10.1007%2Fs00440-004-0406-3/MediaObjects/s00440-004-0406-3flb2.gif, where the rate function H0(t) is a certain asymptotic version of the function https://static-content.springer.com/image/art%3A10.1007%2Fs00440-004-0406-3/MediaObjects/s00440-004-0406-3flb3.gif (case B) or https://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 variablesRandom exponentialsRegular variationExponential Tauberian theoremsCentral limit theoremWeak limit theoremsInfinitely divisible distributionsStable laws
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© Springer-Verlag Berlin Heidelberg 2005