Passage of Lévy Processes across Power Law Boundaries at Small Times

We wish to characterize when a Lévy process $X_{t}$ crosses boundaries like $t^{\kappa}$, κ > 0, in a one- or two-sided sense, for small times t; thus, we inquire when $\text{lim sup}_{t\downarrow 0}\ |X_{t}|/t^{\kappa}$, $\text{lim sup}_{t\downarrow 0}\ X_{t}/t^{\kappa}$ and/or $\text{lim inf}_{t\downarrow 0}\ X_{t}/t^{\kappa}$ are almost surely (a.s.) finite or infinite. Necessary and sufficient conditions are given for these possibilities for all values of κ > 0. This completes and extends a line of research going back to Blumenthal and Getoor in the 1960s. Often (for many values of κ), when the lim sups are finite a.s., they are in fact zero, but the lim sups may in some circumstances take finite, nonzero, values, a.s. In general, the process crosses one- or two-sided boundaries in quite different ways, but surprisingly this is not so for the case κ = 1/2, where a new kind of analogue of an iterated logarithm law with a square root boundary is derived. An integral test is given to distinguish the possibilities in that case.