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Proceedings of the American Mathematical Society
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
Published electronically: October 18, 2007
MathSciNet review: 2350413
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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). $


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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: http://dx.doi.org/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
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.