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Convex hulls of Lévy processes in space time
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Kramer-Bang, David (2023) Convex hulls of Lévy processes in space time. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b3973434
Abstract
In this thesis we apply a stick-breaking representation of the convex minorant and concave majorant of a one-dimensional Lévy process to show multiple probabilistic and geometric properties for the convex hull of a Lévy process. We show a central limit theorem for the fluctuations of the length of the concave majorant of a Lévy process when there is a finite second moment and consider the asymptotic dependence with the extrema of the process itself. The limit fluctuations of the length is also considered in the case where the Lévy process is in the domain of attraction of an α-stable law. In the rest of the thesis we study smoothness properties of the convex hull. Indeed, we characterise a class of Lévy processes whose graph has a continuously differentiable convex hull. Moreover, we also study how smooth the convex hull can be, by studying the growth rate of the convex minorant whenever the right derivative of the convex minorant increases continuously. Lastly, we characteris e the Hölder continuity of the convex hull of a one-dimensional Lévy process.
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Lévy processes, Central limit theorem, Convex functions, Space and time | ||||
Official Date: | August 2023 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Department of Statistics | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Mijatović, Aleksandar | ||||
Format of File: | |||||
Extent: | vii, 178 pages : illustrations | ||||
Language: | eng |
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