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

Serguei Foss (), Takis Konstantopoulos () and Stan Zachary ()
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Serguei Foss: Heriot-Watt University
Takis Konstantopoulos: Heriot-Watt University
Stan Zachary: Heriot-Watt University

Journal of Theoretical Probability, 2007, vol. 20, issue 3, 581-612

Abstract: 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 (search for similar items in EconPapers)
Date: 2007
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Citations: View citations in EconPapers (3)

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DOI: 10.1007/s10959-007-0081-2

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