Passage of levy processes across power law boundaries at small times

• 成果类型: Article
• 署名作者: Bertoin, J.; Doney, R. A.; Maller, R. A.
• 署名单位: Sorbonne Universite; University of Manchester; Australian National University; Australian National University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117907000000097
• 发表日期: 2008
• 页码: 160-197
• 关键词: independent increments; random-walks; STABILITY

摘要

We wish to characterize when a Levy process X, crosses boundaries like t(k), K > 0, in a one- or two-sided sense, for small times t; thus, we inquire when lim sup(t down arrow 0) vertical bar X-t vertical bar/t(k), lim sup(t down arrow 0) X-t/t(k) and/or lim inf(t down arrow 0) X-t/t(k) are almost surely (a.s.) finite or infinite. Necessary and sufficient conditions are given for these possibilities for all values Of K > 0. This completes and extends a line of research going back to Blumenthal and Getoor in the 1960s. Often (for many values Of K), 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 k = 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.