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

DOI:
https://doi.org/10.1090/S0002-9939-08-09163-6

Published electronically:
March 4, 2008

MathSciNet review:
2390532

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Abstract | References | Similar Articles | Additional Information

Abstract: Let denote a Lévy process in with exponent . Taylor (1986) proved that the packing dimension of the range is given by the index

We provide an alternative formulation of in terms of the Lévy exponent . 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.

**[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

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:
https://doi.org/10.1090/S0002-9939-08-09163-6

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.