Small Time One-Sided LIL Behavior for Levy Processes at Zero

We specify a function b(0)(t) in terms of the Levy triplet such that lim sup(t -> 0) X(t)/b(0)(t) is an element of [1, 1.8] a.s. integral(1)(0)(Pi) over bar (+) (b(0)(t)) dt < infinity for any Levy process X with unbounded variation and a Brownian component sigma = 0. We show with an example that there are cases where lim sup(t -> 0)X(t)/b(t) = 1 a.s. but b(t) is not asymptotically equivalent to b(0)(t) as t tends to 0. We achieve this by introducing an integral criterion which checks whether lim sup(t -> 0)X(t)/b(t) is 0, infinity, or a finite positive value for b(t) satisfying very mild conditions and any Levy process.