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ISSN 1088-6826(e) ISSN 0002-9939(p)

Billingsley's packing dimension

Author(s): Manav Das
Journal: Proc. Amer. Math. Soc. 136 (2008), 273-278.
MSC (2000): Primary 28A78, 28A80
Posted: October 18, 2007
MathSciNet review: 2350413
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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:

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2.: P. Billingsley, Hausdorff Dimension in Probability Theory II, Ill. J. Math., 5, 1961, pp. 291-298. MR 0120339 (22:11094)
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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)
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Additional Information:

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

DOI: 10.1090/S0002-9939-07-09069-7
PII: S 0002-9939(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
Posted: October 18, 2007
Communicated by: Jane M. Hawkins
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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