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

 

Singularity and fine fractal properties of a certain class of infinite Bernoulli convolutions with an essential intersection


Authors: M. V. Lebid’ and G. M. Torbin
Translated by: N. Semenov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 87 (2012).
Journal: Theor. Probability and Math. Statist. 87 (2013), 99-115
MSC (2010): Primary 60G30, 11K55, 28A80
Published electronically: March 21, 2014
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove that the probability distribution of a random variable $ \xi $ represented in the form of an infinite series

$\displaystyle \xi = \sum _{k=1}^\infty \xi _k a_k $

is singular, where $ \xi _k$ are independent Bernoulli random variables and where the sequence $ \{a_k\}$ is such that the series $ \sum _{k=1}^\infty a_k$ converges, $ a_k\ge 0$ for all $ k\ge 1$, and, for an arbitrary $ k\in \mathbf {N}$, there exists $ s_{k} \in \mathbf {N}\cup \{0\}$ for which $ s_k>0$ for infinitely many indices $ k$ and $ a_{k} = a_{k+1} = \dots = a_{k+s_{k} } \ge r_{k+s_{k} }$, where $ r_k$ is the tail of the series, namely

$\displaystyle r_k=\sum _{i=k+1}^\infty a_i. $

Under these assumptions, it is shown that the corresponding distribution is a
Bernoulli convolution with essential intersections (that is, almost all with respect to the Hausdorff-Besicovitch dimension points of the spectrum have continuum many different expansions of the form $ \sum _{k=1}^\infty \omega _k a_k$, where $ \omega _k \in \{0,1\}$). Our main attention is paid to the studies of fractal properties of singularly continuous probability measures $ \mu _{\xi }$. In particular, fractal properties of the spectra (minimal closed supports of the above measures) and minimal in the sense of the Hausdorff-Besicovitch dimension dimensional supports of such probability distributions are studied in detail.

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Additional Information

M. V. Lebid’
Affiliation: Department of Mathematical Analysis and Differential Equations, Dragomanov National Pedagogical University, Pirogov Street, 9, Kyiv 01130, Ukraine
Email: mykola.lebid@gmail.com

G. M. Torbin
Affiliation: Department of Mathematical Analysis and Differential Equations, Dragomanov National Pedagogical University, Pirogov Street, 9, Kyiv 01130, Ukraine
Email: torbin7@gmail.com, torbin@iam.uni-bonn.de

DOI: http://dx.doi.org/10.1090/S0094-9000-2014-00907-2
PII: S 0094-9000(2014)00907-2
Keywords: Infinite Bernoulli convolutions, fractals, singularly continuous probability measures, Hausdorff--Besicovitch dimension of a set, Hausdorff dimension of a measure, faithful covering systems
Received by editor(s): April 9, 2012
Published electronically: March 21, 2014
Additional Notes: The research of the first named author was partially supported by the DFG Grant 436 113/97
The research of the second named author was partially supported by the DFG Grants 436 UKR 113/97, DFG KO 1989/6-1, and Humboldt Foundation
Article copyright: © Copyright 2014 American Mathematical Society