Stochastic bounds for Levy processes

2004/10/06 by R. A. Doney
Mathematics · #math.PR #msc:60G51 #msc:60G17

paper · pdf · doi:10.1214/009117904000000315

published as Annals of Probability 2004, Vol. 32, No. 2, 1545-1552 · Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/009117904000000315

arxiv created 2004/10/06 · arxiv updated 2009/12/01

Abstract

Using the Wiener-Hopf factorization, it is shown that it is possible to bound the path of an arbitrary Levy process above and below by the paths of two random walks. These walks have the same step distribution, but different random starting points. In principle, this allows one to deduce Levy process versions of many known results about the large-time behavior of random walks. This is illustrated by establishing a comprehensive theorem about Levy processes which converge to ∞ in probability.

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