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

MathSciNet review:
3241449

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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

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 , where ). Our main attention is paid to the studies of fractal properties of singularly continuous probability measures . 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:
https://doi.org/10.1090/S0094-9000-2014-00907-2

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