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Theory of Probability and Mathematical Statistics
Theory of Probability and Mathematical Statistics
ISSN 1547-7363(online) ISSN 0094-9000(print)

Fractal properties of some Bernoulli convolutions


Authors: Ya. V. Goncharenko, M. V. Pratsyovytyĭ and G. M. Torbin
Translated by: N. Semenov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 79 (2008).
Journal: Theor. Probability and Math. Statist. 79 (2009), 39-55
MSC (2000): Primary 60G30, 11K55, 28A80
Published electronically: December 29, 2009
MathSciNet review: 2494534
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Abstract: We study the structure and topological, metric, and fractal properties of the distribution of the random variable

$\displaystyle \xi=\sum_{k=1}^\infty \xi_k a_k, $

where $ \sum_{k=1}^\infty a_k$ is a convergent series of positive terms $ a_k$ such that

$\displaystyle a_{3k-2}= a_{3k-1}+a_{3k}, $

$ a_{i} \ge\nobreak a_{i+1}+a_{i+2}+\cdots$, $ i \ne 3k-2$, $ k \in\mathbf N$, and where $ \xi_k$ are independent random variables assuming two values 0 and $ 1$ with the probabilities $ p_{0k}$ and $ p_{1k}$, respectively. We prove that the distribution of $ \xi$ is either purely discrete or purely singularly continuous. We obtain the criteria for a distribution to belong to each of these types. The topological-metric structure of the distribution is studied in the continuous case. The main result of the paper describes the fractal properties of the distribution of the random variable $ \xi$. The relations are obtained for the Hausdorff-Besicovitch spectrum (the minimal closed support) of the random variable $ \xi$ and for the Hausdorff dimension of the corresponding probability measure $ \mu_{\xi}$.


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Additional Information

Ya. V. Goncharenko
Affiliation: Department of Higher Mathematics, National Dragomanov Pedagogical University, Pirogova Street 9, Kyiv 01130, Ukraine
Email: yan_a@ukr.net

M. V. Pratsyovytyĭ
Affiliation: Department of Higher Mathematics, National Dragomanov Pedagogical University, Pirogova Street 9, Kyiv 01130, Ukraine
Email: prats4@yandex.ru

G. M. Torbin
Affiliation: Department of Higher Mathematics, National Dragomanov Pedagogical University, Pirogova Street 9, Kyiv 01130, Ukraine
Email: torbin@imath.kiev.ua

DOI: http://dx.doi.org/10.1090/S0094-9000-09-00779-0
PII: S 0094-9000(09)00779-0
Keywords: Bernoulli convolution, singularly continuous probability distribution, Hausdorff--Besicovitch dimension, Hausdorff dimension of a distribution (measure), the set of incomplete sums of a series
Received by editor(s): November 27, 2007
Published electronically: December 29, 2009
Additional Notes: The first author is supported by the grant DFG 436 113/80
The second author is supported by the grants DFG 436 UKR 113/78 and DFG 436 113/80
The third author is supported by the grants DFG 436 UKR 113/78 and DFG 436 113/80 and by the Alexander von Humboldt Foundation
Article copyright: © Copyright 2009 American Mathematical Society