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Journal of Theoretical Probability

, Volume 20, Issue 3, pp 581–612 | Cite as

Discrete and Continuous Time Modulated Random Walks with Heavy-Tailed Increments

  • Serguei Foss
  • Takis Konstantopoulos
  • Stan Zachary
Article

Abstract

We consider a modulated process S which, conditional on a background process X, has independent increments. Assuming that S drifts to −∞ and that its increments (jumps) are heavy-tailed (in a sense made precise in the paper), we exhibit natural conditions under which the asymptotics of the tail distribution of the overall maximum of S can be computed. We present results in discrete and in continuous time. In particular, in the absence of modulation, the process S in continuous time reduces to a Lévy process with heavy-tailed Lévy measure. A central point of the paper is that we make full use of the so-called “principle of a single big jump” in order to obtain both upper and lower bounds. Thus, the proofs are entirely probabilistic. The paper is motivated by queueing and Lévy stochastic networks.

Keywords

Random walk Subexponential distribution Heavy tails Pakes-Veraverbeke theorem Processes with independent increments Regenerative process 

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Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Serguei Foss
    • 1
  • Takis Konstantopoulos
    • 1
  • Stan Zachary
    • 1
  1. 1.Department of Actuarial Mathematics and Statistics, School of Mathematical SciencesHeriot-Watt UniversityEdinburghUK

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