Stochastic bounds for Lévy processes

Using the Wiener–Hopf factorization, it is shown that it is possible to bound the path of an arbitrary Lévy 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 Lévy process versions of many known results about the large-time behavior of random walks. This is illustrated by establishing a comprehensive theorem about Lévy processes which converge to ∞ in probability. 1. Introduction. If X = (Xt, t ≥ 0) is an arbitrary Lévy process, we would frequently like to be able to assert that some aspect of its behavior as t → ∞ can be seen to be true “by analogy with known results for random walks. ” An obvious way to try to justify such a claim is via the random walk S(δ): = (X(nδ), n ≥ 0), for fixed δ> 0. (This process is often called the δ-skeleton of X.) However, it i