Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-27T02:09:06.490Z Has data issue: false hasContentIssue false
  • English
  • Français

Packing-dimension profiles and fractional Brownian motion†

Published online by Cambridge University Press:  01 July 2008

DAVAR KHOSHNEVISAN  and
YIMIN XIAO
DAVAR KHOSHNEVISAN
Affiliation:
Department of Mathematics, 155 S. 1400 E., JWB 233, University of Utah, Salt Lake City, UT 84112–0090, U.S.A. e-mail: [email protected] URL: http://www.math.utah.edu/~davar
YIMIN XIAO
Affiliation:
Department of Statistics and Probability, A-413 Wells Hall, Michigan State University, East Lansing, MI 48824, U.S.A. e-mail: [email protected] URL: http://www.stt.msu.edu/~xiaoyimi
Get access Rights & Permissions [Opens in a new window]

Abstract

In order to compute the packing dimension of orthogonal projections Falconer and Howroyd [3] have introduced a family of packing dimension profiles Dims that are parametrized by real numbers s > 0. Subsequently, Howroyd [5] introduced alternate s-dimensional packing dimension profiles P-Dims by using Caratheodory-type packing measures, and proved, among many other things, that P-DimsE = DimsE for all integers s > 0 and all analytic sets E ⊆ RN.

The aim of this paper is to prove that P-DimsE = DimsE for all real numbers s > 0 and analytic sets E ⊆ RN. This answers a question of Howroyd [5, p. 159]. Our proof hinges on establishing a new property of fractional Brownian motion.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 145 , Issue 1 , July 2008 , pp. 205 - 213
Copyright
Copyright © Cambridge Philosophical Society 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Falconer, K. J.. Fractal Geometry—Mathematical Foundations And Applications (Wiley & Sons Ltd. 1990).CrossRefGoogle Scholar
[2]Falconer, K. J. and Howroyd, J. D.. Projection theorems for box and packing dimensions. Math. Proc. Camb. Phil. Soc. 119 (1996), 287–295.CrossRefGoogle Scholar
[3]Falconer, K. J. and Howroyd, J. D.. Packing dimensions for projections and dimension profiles. Math. Proc. Camb. Phil. Soc. 121 (1997), 269–286.CrossRefGoogle Scholar
[4]Falconer, K. J.. and Mattila, P.. The packing dimension of projections and sections of measures. Math. Proc. Camb. Phil. Soc. 119 (1996), 695–713.CrossRefGoogle Scholar
[5]Howroyd, J. D.. Box and packing dimensions of projections and dimension profiles. Math. Proc. Camb. Phil. Soc. 130 (2001), 135–160.CrossRefGoogle Scholar
[6]Hu, X. and Taylor, S. J.. Fractal properties of products and projections of measures in d. Math Proc. Camb. Phil. Soc. 115 (1994), 527–544.CrossRefGoogle Scholar
[7]Järvenpää, M.. On the upper Minkowski dimension, the packing dimension, and othogonal projections. Annales Acad. Sci. Fenn. A Dissertat. 99 (1994), 34 pp.Google Scholar
[8]Kahane, J.-P.. Some Random Series of Functions. 2nd edition (Cambridge University Press, 1985).Google Scholar
[9]Mattila, P.. Geometry of Sets and Measures in Euclidean Spaces (Cambridge University Press, 1995).CrossRefGoogle Scholar
[10]Talagrand, M. and Xiao, Y.. Fractional Brownian motion and packing dimension. J. Theoret. Probab. 9 (1996), 579–593.CrossRefGoogle Scholar
[11]Taylor, S. J. and Tricot, C.. Packing measure and its evaluation for a Brownian path. Trans. Amer. Math. Soc. 288 (1985), 679–699.CrossRefGoogle Scholar
[12]Tricot, C.. Two definitions of fractional dimension. Math. Proc. Camb. Phil. Soc. 91 (1982), 57–74.CrossRefGoogle Scholar
[13]Xiao, Y.. Packing dimension of the image of fractional Brownian motion. Statist. Probab. Lett. 33 (1997), 379–387.CrossRefGoogle Scholar