Packing dimension of the range of a Lévy process
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- by Davar Khoshnevisan and Yimin Xiao
- Proc. Amer. Math. Soc. 136 (2008), 2597-2607
- DOI: https://doi.org/10.1090/S0002-9939-08-09163-6
- Published electronically: March 4, 2008
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Abstract:
Let $\{X(t)\}_{t\ge 0}$ denote a Lévy process in ${\mathbf {R}}^d$ with exponent $\Psi$. Taylor (1986) proved that the packing dimension of the range $X([0 ,1])$ is given by the index \begin{equation*} {(0.1)}\qquad \qquad \gamma ’ = \sup \left \{\alpha \ge 0: \liminf _{r \to 0^+} \int _0^1 \frac {\mathrm {P} \left \{|X(t)| \le r\right \}}{r^\alpha } dt =0\right \}.\qquad \qquad \end{equation*} We provide an alternative formulation of $\gamma ’$ in terms of the Lévy exponent $\Psi$. Our formulation, as well as methods, are Fourier-analytic, and rely on the properties of the Cauchy transform. We show, through examples, some applications of our formula.References
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Bibliographic Information
- Davar Khoshnevisan
- Affiliation: Department of Mathematics, The University of Utah, 155 S. 1400 East, Salt Lake City, Utah 84112–0090
- MR Author ID: 302544
- Email: davar@math.utah.edu
- Yimin Xiao
- Affiliation: Department of Statistics and Probability, A-413 Wells Hall, Michigan State University, East Lansing, Michigan 48824
- Email: xiao@stt.msu.edu
- Received by editor(s): June 21, 2006
- Received by editor(s) in revised form: January 25, 2007, and March 1, 2007
- Published electronically: March 4, 2008
- Additional Notes: This research was partially supported by a grant from the National Science Foundation
- Communicated by: Richard C. Bradley
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 2597-2607
- MSC (2000): Primary 60J30, 60G17, 28A80
- DOI: https://doi.org/10.1090/S0002-9939-08-09163-6
- MathSciNet review: 2390532