Discrete and continuous time modulated random walks with heavy-tailed increments

Serguei Foss, Panagiotis Takis Konstantopoulos, Stan Zachary

Research output: Contribution to journalArticle

30 Citations (Scopus)

Abstract

We consider a modulated process S which, conditional on a background process X, has independent increments. Assuming that S drifts to -8 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. © 2007 Springer Science+Business Media, LLC.

Original languageEnglish
Pages (from-to)581-612
Number of pages32
JournalJournal of Theoretical Probability
Volume20
Issue number3
DOIs
Publication statusPublished - Sep 2007

Keywords

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

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