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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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  • [1] T. W. Anderson, The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities, Proc. Amer. Math. Soc. 6 (1955), 170-176. MR 0069229 (16:1005a)
  • [2] Jean Bertoin, Lévy Processes, Cambridge University Press, Cambridge, 1998. MR 1406564 (98e:60117)
  • [3] Jean Bertoin, Subordinators: Examples and Applications, Lectures on Probability Theory and Statistics (Saint-Flour, 1997), 1999, pp. 1-91. MR 1746300 (2002a:60001)
  • [4] R. M. Blumenthal and R. K. Getoor, Sample functions of stochastic processes with stationary independent increments, J. Math. Mech. 10 (1961), 493-516. MR 0123362 (23:A689)
  • [5] Salomon Bochner, Harmonic Analysis and the Theory of Probability, University of California Press, Berkeley and Los Angeles, 1955. MR 0072370 (17:273d)
  • [6] Richard Durrett, Probability: Theory and Examples, 2nd edition, Duxbury Press, Belmont, CA, 1996. MR 1609153 (98m:60001)
  • [7] B. E. Fristedt and S. James Taylor, The packing measure of a subordinator, Probab. Th. Rel. Fields 92 (1992), 493-510. MR 1169016 (93e:60150)
  • [8] W. J. Hendricks, A uniform lower bound for Hausdorff dimension for transient symmetric Lévy processes, Ann. Probab. 11 (3) (1983), 589-592. MR 704545 (85a:60043)
  • [9] J. P. Holmes, W. N. Hudson, and J. David Mason, Operator stable laws: Multiple exponents and elliptical symmetry, Ann. Probab. 10 (1982), 602-612. MR 659531 (83i:60012)
  • [10] W. N. Hudson and J. David Mason, Operator-self-similar processes in a finite-dimensional space, Trans. Amer. Math. Soc. 273 (1982), 281-297. MR 664042 (83h:60051)
  • [11] Harry Kesten, Hitting probabilities of single points for processes with stationary independent increments, Memoirs of the American Mathematical Society 93, American Mathematical Society, Providence, RI, 1969. MR 0272059 (42:6940)
  • [12] Davar Khoshnevisan and Yimin Xiao, Weak unimodality of finite measures, and an application to potential theory of additive Lévy processes, Proc. Amer. Math. Soc. 131 (8) (2003), 2611-2616 (electronic). MR 1974662 (2004h:60016)
  • [13] Davar Khoshnevisan, Yimin Xiao, and Yuquan Zhong, Measuring the range of an additive Lévy process, Ann. Probab. 31 (2) (2003), 1097-1141. MR 1964960 (2004c:60155)
  • [14] Mark M. Meerschaert and H.-P. Scheffler, Limit Distributions for Sums of Independent Random Vectors, Wiley & Sons, New York, 2001. MR 1840531 (2002i:60047)
  • [15] Mark M. Meerschaert and Yimin Xiao, Dimension results for sample paths of operator stable Lévy processes, Stoch. Proc. Appl. 115 (2005), 55-75. MR 2105369 (2005k:60238)
  • [16] Steven Orey, Polar sets for processes with stationary independent increments, Markov Processes and Potential Theory (Proc. Sympos. Math. Res. Center, Madison, Wis., 1967), Wiley, New York, 1967, pp. 117-126. MR 0229305 (37:4879)
  • [17] William E. Pruitt, The Hausdorff dimension of the range of a process with stationary independent increments, J. Math. Mech. 19 (1969), 371-378. MR 0247673 (40:936)
  • [18] William E. Pruitt and S. James Taylor, Packing and covering indices for a general Lévy process, Ann. Probab. 24 (2) (1996), 971-986. MR 1404539 (97g:60096)
  • [19] William E. Pruitt and S. James Taylor, Sample path properties of processes with stable components, Z. Wahrsch. Verw. Gebiete 12 (1969), 267-289. MR 0258126 (41:2773)
  • [20] Ken-iti Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge, 1999. Translated from the 1990 Japanese original; revised by the author. MR 1739520 (2003b:60064)
  • [21] Michael Sharpe, Operator-stable probability distributions on vector groups, Trans. Amer. Math. Soc. 136 (1969), 51-65. MR 0238365 (38:6641)
  • [22] Dennis Sullivan, Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups, Acta Math. 153 (3-4) (1984), 259-277. MR 766265 (86c:58093)
  • [23] S. James Taylor, The use of packing measure in the analysis of random sets, Stochastic Processes and Their Applications (Nagoya, 1985), Springer, Berlin, 1986, pp. 214-222. MR 872112 (88i:60078)
  • [24] Claude Tricot Jr., Two definitions of fractional dimension, Math. Proc. Cambridge Philos. Soc. 91 (1) (1982), 57-74. MR 633256 (84d:28013)
  • [25] Yimin Xiao, Random fractals and Markov processes, In: Fractal Geometry and Applications (Michel L. Lapidus and Machiel van Frankenhuijsen, editors), American Mathematical Society, Providence, RI, 2004, pp. 261-338. MR 2112126 (2006a:60065)

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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

Yimin Xiao
Affiliation: Department of Statistics and Probability, A-413 Wells Hall, Michigan State University, East Lansing, Michigan 48824

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.

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