The principle of a single big jump: discrete and continuous time modulated random walks with heavy-tailed increments

Details The principle of a single big jump: discrete and continuous time modulated random walks with heavy-tailed increments The principle of a single big jump: discrete and continuous time modulated random walks with heavy-tailed increments

2005-09-26

arxiv.org/abs/math/0509605v1

Journal of Theoretical Probability, 20 (2007), No.3, 581-612

Mathematics

Probability

29 pages

Scientific paper

We consider a modulated process S which, conditional on a background process X, has independent increments. Assuming that S drifts to -infinity 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 Levy process with heavy-tailed Levy 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 Levy stochastic networks.

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