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Probability Theory and Related Fields

, Volume 82, Issue 2, pp 259–269 | Cite as

Subexponential distributions and characterizations of related classes

  • Claudia Klüppelberg
Article

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.

Keywords

Stochastic Process Asymptotic Behaviour Probability Theory Service Time Time Distribution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • Claudia Klüppelberg
    • 1
  1. 1.Seminar für StatistikUniversität MannheimMannheimFederal Republic of Germany

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