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Subexponential distributions and characterizations of related classes
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  • Published: June 1989

Subexponential distributions and characterizations of related classes

  • Claudia Klüppelberg1 

Probability Theory and Related Fields volume 82, pages 259–269 (1989)Cite this article

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Summary

LetL(γ),γ≧0, γ≧0, denote the class of distributionsF satisfying

$$\mathop {\lim }\limits_{x \to \infty } \bar F^{2*} (x)/\bar F(x) = 2\mathop \smallint \limits_0^\infty e^{\gamma y} dF(y)< \infty$$
((i))
$$\mathop {\lim }\limits_{x \to \infty } \bar F^{2*} (x - y)/\bar F(x) = e^{\gamma y} \forall y \in \mathbb{R}.$$
((ii))

The classesL(γ), for γ>0, are characterized by means of subexponential densities. As an application we derive a result on the asymptotic behaviour of densities of random sums. In particular for anM/G/1 queue, we relate the tail behaviour of the stationary waiting time density to that of the service time distribution.

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Authors and Affiliations

  1. Seminar für Statistik, Universität Mannheim, D-6800, Mannheim, Federal Republic of Germany

    Claudia Klüppelberg

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  1. Claudia Klüppelberg
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Klüppelberg, C. Subexponential distributions and characterizations of related classes. Probab. Th. Rel. Fields 82, 259–269 (1989). https://doi.org/10.1007/BF00354763

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  • Received: 30 June 1987

  • Issue Date: June 1989

  • DOI: https://doi.org/10.1007/BF00354763

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Keywords

  • Stochastic Process
  • Asymptotic Behaviour
  • Probability Theory
  • Service Time
  • Time Distribution
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