Open Access
January 2008 Passage of Lévy processes across power law boundaries at small times
J. Bertoin, R. A. Doney, R. A. Maller
Ann. Probab. 36(1): 160-197 (January 2008). DOI: 10.1214/009117907000000097


We wish to characterize when a Lévy process Xt crosses boundaries like tκ, κ>0, in a one- or two-sided sense, for small times t; thus, we inquire when lim sup t↓0|Xt|/tκ, lim sup t↓0Xt/tκ and/or lim inf t↓0Xt/tκ 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.


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J. Bertoin. R. A. Doney. R. A. Maller. "Passage of Lévy processes across power law boundaries at small times." Ann. Probab. 36 (1) 160 - 197, January 2008.


Published: January 2008
First available in Project Euclid: 28 November 2007

zbMATH: 1140.60025
MathSciNet: MR2370602
Digital Object Identifier: 10.1214/009117907000000097

Primary: 60F15 , 60J30
Secondary: 60G10 , 60G17 , 60J65

Keywords: crossing power law boundaries , Lévy processes , limiting and lim sup behavior

Rights: Copyright © 2008 Institute of Mathematical Statistics

Vol.36 • No. 1 • January 2008
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