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Packing-dimension profiles and fractional Brownian motion

Published online by Cambridge University Press:  01 July 2008

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: davar@math.utah.edu 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: xiao@stt.msu.edu URL: http://www.stt.msu.edu/~xiaoyimi

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 ERN.

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

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2008

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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), 287295.CrossRefGoogle Scholar
[3]Falconer, K. J. and Howroyd, J. D.. Packing dimensions for projections and dimension profiles. Math. Proc. Camb. Phil. Soc. 121 (1997), 269286.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), 695713.CrossRefGoogle Scholar
[5]Howroyd, J. D.. Box and packing dimensions of projections and dimension profiles. Math. Proc. Camb. Phil. Soc. 130 (2001), 135160.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), 527544.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), 579593.CrossRefGoogle Scholar
[11]Taylor, S. J. and Tricot, C.. Packing measure and its evaluation for a Brownian path. Trans. Amer. Math. Soc. 288 (1985), 679699.CrossRefGoogle Scholar
[12]Tricot, C.. Two definitions of fractional dimension. Math. Proc. Camb. Phil. Soc. 91 (1982), 5774.CrossRefGoogle Scholar
[13]Xiao, Y.. Packing dimension of the image of fractional Brownian motion. Statist. Probab. Lett. 33 (1997), 379387.CrossRefGoogle Scholar