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

We specify a function b0(t) in terms of the Lévy triplet such that lim supt→0Xt/b0(t) ∈ [1, 1.8] a.s. iff ∫ 1 0 Π (+)(b0(t))dt < ∞, for any Lévy process X with unbounded variation and Brownian component σ = 0. We show with example that there are cases when lim supt→0Xt/b(t) = 1 a.s., but b(t) is not asymptotically equivalent to b0(t), as t goes to 0. We achieve this by introducing an integral criteria, which checks whether lim supt→0Xt/b(t) is 0, infinity or a finite positive value, for b(t) satisfying very mild conditions and any Lévy process.