PASSAGE OF LÉVY PROCESSES ACROSS POWER LAW BOUNDARIES AT SMALL TIMES
Identifieur interne : 008B47 ( Main/Exploration ); précédent : 008B46; suivant : 008B48PASSAGE OF LÉVY PROCESSES ACROSS POWER LAW BOUNDARIES AT SMALL TIMES
Auteurs : J. Bertoin [France, Royaume-Uni, Australie] ; R. A. Doney ; R. A. MallerSource :
- Annals of probability [ 0091-1798 ] ; 2008.
Descripteurs français
- Pascal (Inist)
English descriptors
- KwdEn :
Abstract
We wish to characterize when a Lévy process Xt crosses boundaries like tK, K > 0, in a one- or two-sided sense, for small times t; thus, we inquire when lim supt ↓0 |Xt|/tK, lim supt ↓0 Xt/tK and/or lim inft ↓0 Xt/tK 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 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 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.
Affiliations:
- Australie, France, Royaume-Uni
- Angleterre, Grand Manchester, Île-de-France
- Manchester, Paris
- Université Pierre-et-Marie-Curie, Université de Manchester
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A." last="Doney">R. A. Doney</name></author><author><name sortKey="Maller, R A" sort="Maller, R A" uniqKey="Maller R" first="R. A." last="Maller">R. A. Maller</name></author></analytic><series><title level="j" type="main">Annals of probability</title><title level="j" type="abbreviated">Ann. probab.</title><idno type="ISSN">0091-1798</idno><imprint><date when="2008">2008</date></imprint></series></biblStruct></sourceDesc><seriesStmt><title level="j" type="main">Annals of probability</title><title level="j" type="abbreviated">Ann. probab.</title><idno type="ISSN">0091-1798</idno></seriesStmt></fileDesc><profileDesc><textClass><keywords scheme="KwdEn" xml:lang="en"><term>Integral</term><term>Law of iterated logarithm</term><term>Lévy process</term><term>Necessary and sufficient condition</term><term>Power law</term><term>Probability theory</term></keywords><keywords scheme="Pascal" xml:lang="fr"><term>Processus Lévy</term><term>Loi puissance</term><term>Condition nécessaire suffisante</term><term>Loi logarithme itéré</term><term>Intégrale</term><term>Théorie probabilité</term></keywords></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">We wish to characterize when a Lévy process X<sub>t</sub> crosses boundaries like t<sup>K</sup>, K > 0, in a one- or two-sided sense, for small times t; thus, we inquire when lim sup<sub>t </sub>↓<sub>0</sub> |X<sub>t</sub>|/t<sup>K</sup>, lim sup<sub>t </sub>↓<sub>0</sub> X<sub>t</sub>/t<sup>K</sup> and/or lim inf<sub>t </sub>↓<sub>0</sub> X<sub>t</sub>/t<sup>K</sup> 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 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 K = 1/2, where a new kind of analogue of an iterated logarithm law with a square root boundary is derived. 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