Fractal properties of random variables with independent -symbols

Authors:
R. O. Nikiforov and G. M. Torbin

Translated by:
S. V. Kvasko

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **86** (2012).

Journal:
Theor. Probability and Math. Statist. **86** (2013), 169-182

MSC (2010):
Primary 60G30, 11K55, 28A80

DOI:
https://doi.org/10.1090/S0094-9000-2013-00896-5

Published electronically:
August 20, 2013

MathSciNet review:
2986457

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study an equivalent definition of the Hausdorff-Besicovitch dimension in terms of a system of cylinders of the -expansion. Sufficient conditions for the system to be faithful for the evaluation of the Hausdorff-Besicovitch dimension in the unit interval are found; fine fractal properties of probability measures with independent -digits are investigated (we do not assume that the -digits are identically distributed).

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Additional Information

**R. O. Nikiforov**

Affiliation:
Department of Higher Mathematics, Dragomanov National Pedagogical University, Pirogova Street 9, Kyiv 01130, Ukraine

Email:
rnikiforov@gmail.com

**G. M. Torbin**

Affiliation:
Department of Mathematical Analysis and Differential Equations, Dragomanov National Pedagogical University, Pirogova Street 9, Kyiv 01130, Ukraine – and – Department of Fractal Analysis, Institute for Mathematics, National Academy of Science of Ukraine, Tereshchenkivs’ka Street 3, Kyiv 01130, Ukraine

Email:
torbin7@gmail.com, torbin@imath.kiev.ua

DOI:
https://doi.org/10.1090/S0094-9000-2013-00896-5

Keywords:
$Q_{\infty}$-expansions,
faithful systems of coverings,
singularly continuous probability distributions,
Hausdorff--Besicovitch dimension of a set,
Hausdorff dimension of a measure

Received by editor(s):
November 30, 2011

Published electronically:
August 20, 2013

Additional Notes:
The first author was supported by the Project DFG 436113/97

The second author was supported by the Projects DFG 436 UKR 113/97 and DFG KO 1989/6-1 and the Alexander von Humboldt Foundation

Article copyright:
© Copyright 2013
American Mathematical Society