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Theory of Probability and Mathematical Statistics

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Fractal properties of random variables with independent $ Q_{\infty}$-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
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Abstract: We study an equivalent definition of the Hausdorff-Besicovitch dimension in terms of a system $ \Phi (Q_\infty )$ of cylinders of the $ Q_\infty $-expansion. Sufficient conditions for the system $ \Phi (Q_\infty )$ 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 $ Q_\infty $-digits are investigated (we do not assume that the $ Q_\infty $-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