Stability of the overshoot for Lévy processes

We give equivalences for conditions like X(T(r))/r → 1 and X(T*(r))/r → 1, where the convergence is in probability or almost sure, both as r → 0 and r → ∞, where X is a Lévy process and T(r) and T*(r) are the first exit times of X out of the strip {(t, y): t > 0, |Y| ≤ r} and half-plane {(t, y): t > 0, y ≤ r}, respectively. We also show, using a result of Kesten, that X(T*(r))/r → 1 a.s. as r → 0 is equivalent to X "creeping" across a level.