Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Billingsley's packing dimension


Author: Manav Das
Journal: Proc. Amer. Math. Soc. 136 (2008), 273-278
MSC (2000): Primary 28A78, 28A80
DOI: https://doi.org/10.1090/S0002-9939-07-09069-7
Published electronically: October 18, 2007
MathSciNet review: 2350413
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For a stochastic process on a finite state space, we define the notion of a packing measure based on the naturally defined cylinder sets. For any two measures $ \nu$, $ \gamma$, corresponding to the same stochastic process, if

$\displaystyle F \subseteq \left \{ \omega \in \Omega : \lim_{n} \frac{\log \gamma(c_{n}(\omega))}{\log \nu(c_{n}(\omega))} = \theta \right \}, $

then we prove that

$\displaystyle {\rm {Dim}}_{\nu}(F) = \theta ~{\rm {Dim}}_{\gamma}(F). $


References [Enhancements On Off] (What's this?)

  • 1. P. Billingsley, Hausdorff Dimension in Probability Theory I, Ill. J. Math., 4, 1960, pp. 187-209. MR 0131903 (24:A1750)
  • 2. P. Billingsley, Hausdorff Dimension in Probability Theory II, Ill. J. Math., 5, 1961, pp. 291-298. MR 0120339 (22:11094)
  • 3. Helmut Cajar, Billingsley dimension in probability spaces, Lecture Notes in Mathematics, 892, Springer-Verlag, Berlin-New York, 1981. MR 654147 (84a:10055)
  • 4. C. D. Cutler, A note on equivalent interval covering systems for Hausdorff dimension on $ \mathbf{R}$, Int. J. Math. Math. Sci., 11, no. 4, 1988, pp. 643-650. MR 959443 (89h:28008)
  • 5. Chao Shou Dai and S. James Taylor, Defining fractals in a probability space, Illinois J. Math., 38, no. 3, 1994, pp. 480-500. MR 1269700 (95f:28011)
  • 6. Manabendra Das, Pointwise Local Dimensions, Ph. D. Thesis, The Ohio State University, 1996.
  • 7. Manav Das, Packing Measures, Dimensions and Mutual Singularity, preprint.
  • 8. Manav Das, Hausdorff Measures, Dimensions and Mutual Singularity, Trans. Amer. Math. Soc, 357, no. 11, 2005, pp. 4249-4268 MR 2156710 (2006g:28010)
  • 9. G. A. Edgar, Measure, topology, and fractal geometry, Springer-Verlag, New York, 1990. MR 1065392 (92a:54001)
  • 10. S. James Taylor, The measure theory of random fractals, Math. Proc. Cambridge Philos. Soc., 100, no. 3, 1986, pp. 383-406. MR 857718 (87k:60189)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 28A78, 28A80

Retrieve articles in all journals with MSC (2000): 28A78, 28A80


Additional Information

Manav Das
Affiliation: Department of Mathematics, 328 Natural Sciences Building, University of Louisville, Louisville, Kentucky 40292
Email: manav@louisville.edu

DOI: https://doi.org/10.1090/S0002-9939-07-09069-7
Keywords: Billingsley's dimension, packing dimension, Hausdorff dimension
Received by editor(s): May 4, 2006
Received by editor(s) in revised form: December 18, 2006
Published electronically: October 18, 2007
Communicated by: Jane M. Hawkins
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society