Passage of Lévy Processes across Power Law Boundaries at Small Times

• Published in: The Annals of probability 2008-01, Vol.36 (1), p.160-197
• Main Authors: Bertoin, J., Doney, R. A., Maller, R. A.
• Format: Article
• Language: English
• Subjects: 60F15; 60G10; 60G17; 60J30; 60J65; Approximation; Chebyshevs inequality; crossing power law boundaries; Exact sciences and technology; General topics; Inverse functions; limiting and lim sup behavior; Logarithms; Lévy processes; Mathematical models; Mathematics; Power laws; Probability; Probability and statistics; Probability theory and stochastic processes; Random variables; Sciences and techniques of general use; Series convergence; Studies; Sufficient conditions; Urelements
• ISSN: 0091-1798; 2168-894X
• DOI: 10.1214/009117907000000097

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.