Small time two-sided LIL behavior for Lévy processes at zero

Abstract

We specify a function b0(t) in terms of the Levy triplet such that lim sup t→0Xt/b0(t)∈[1,1.8] a.s. iff \(\int_{0}^{1}\overline{ \varPi }^{(+)}(b_{0}(t))\,dt<\infty\) for any Levy process X with unbounded variation and a Brownian component σ=0. We show with an example that there are cases where lim sup t→0Xt/b(t)=1 a.s. but b(t) is not asymptotically equivalent to b0(t) as t tends to 0. We achieve this by introducing an integral criterion which checks whether lim sup t→0Xt/b(t) is 0, infinity, or a finite positive value for b(t) satisfying very mild conditions and any Levy process.