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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Packing dimension of the range of a Lévy process


Authors: Davar Khoshnevisan and Yimin Xiao
Journal: Proc. Amer. Math. Soc. 136 (2008), 2597-2607
MSC (2000): Primary 60J30, 60G17, 28A80
Published electronically: March 4, 2008
MathSciNet review: 2390532
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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

$\displaystyle {(0.1)}\qquad\qquad \gamma' = \sup\left\{\alpha\ge 0: \liminf_{r ... ... \left\{\vert X(t)\vert \le r\right\}}{r^\alpha} \, dt =0\right\}.\qquad\qquad $

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.


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Additional Information

Davar Khoshnevisan
Affiliation: Department of Mathematics, The University of Utah, 155 S. 1400 East, Salt Lake City, Utah 84112–0090
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

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09163-6
PII: S 0002-9939(08)09163-6
Keywords: L\'evy processes, operator stable L\'evy processes, packing dimension, Hausdorff dimension.
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
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.