Queueing Systems

, Volume 46, Issue 1–2, pp 15–33 | Cite as

Tail Asymptotics for the Supremum of a Random Walk when the Mean Is not Finite

  • Denis Denisov
  • Serguei Foss
  • Dima Korshunov
Article

Abstract

We consider the sums S n 1+⋯+ξ n of independent identically distributed random variables. We do not assume that the ξ's have a finite mean. Under subexponential type conditions on distribution of the summands, we find the asymptotics of the probability P{M>x} as x→∞, provided that M=sup {S n ,n≥1} is a proper random variable. Special attention is paid to the case of tails which are regularly varying at infinity. We provide some sufficient conditions for the integrated weighted tail distribution to be subexponential. We supplement these conditions by a number of examples which cover both the infinite- and the finite-mean cases. In particular, we show that the subexponentiality of distribution F does not imply the subexponentiality of its integrated tail distribution FI.

supremum of sums of random variables large deviation probabilities subexponential distribution integrated weighted tail distribution 

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Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Denis Denisov
    • 1
  • Serguei Foss
    • 1
  • Dima Korshunov
    • 1
  1. 1.Department of Actuarial Mathematics and Statistics, School of Mathematical and Computer SiencesHeriot-Watt UniversityEdinburghScotland

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