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), 99115
MSC (2010):
Primary 60G30, 11K55, 28A80
Published electronically:
March 21, 2014
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References 
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Additional Information
Abstract: We prove that the probability distribution of a random variable represented in the form of an infinite series is singular, where are independent Bernoulli random variables and where the sequence is such that the series converges, for all , and, for an arbitrary , there exists for which for infinitely many indices and , where is the tail of the series, namely 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 HausdorffBesicovitch 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 HausdorffBesicovitch dimension dimensional supports of such probability distributions are studied in detail.
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 S. Albeverio and G. Torbin, On fine fractal properties of generalized infinite Bernoulli convolutions, Bull. Sci. Math. 132 (2008), no. 8, 711727. MR 2474489 (2010e:28006)
 2.
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 S. Albeverio and G. Torbin, Image measures of infinite product measures and generalized Bernoulli convolutions, Proceedings of Dragomanov National Pedagogical University, ser. 1, Physics and Mathematics 5 (2004), 248264.
 4.
 S. Albeverio and G. Torbin, Fractal properties of singularly continuous probability distributions with independent digits, Bull. Sci. Math. 129 (2005), no. 4, 356367. MR 2134126 (2006b:28013)
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 P. Erdős, On a family of symmetric Bernoulli convolutions, Amer. J. Math. 61 (1939), 974975. MR 0000311 (1:52a)
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 K. J. Falconer, Fractal Geometry, John Wiley & Sons, 1990. MR 1102677 (92j:28008)
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 A. M. Garsia, Arithmetic properties of Bernoulli convolutions, Trans. Amer. Math. Soc. 102 (1962), 409432. MR 0137961 (25:1409)
 11.
 Ya. Gontcharenko, M. Pratsiovytyi, and G. Torbin, On fractal properties of some Bernoulli convolutions, Prob. Theory and Math. Statist. 79 (2009), 3955. MR 2494534 (2010d:60101)
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 13.
 P. Lévy, Sur les séries dont les termes sont des variables indépendantes, Studia Math. 3 (1931), 119155.
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 Y. Peres, W. Schlag, and B. Solomyak, Sixty years of Bernoulli convolutions, Fractal Geometry and Stochastics II, Progress in Probab., vol. 46, 2000, pp. 3965. MR 1785620 (2001m:42020)
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 B. Solomyak, On the random series (an Erdős problem), Ann. Math. 142 (1995), 611625. MR 1356783 (97d:11125)
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 M. V. Pratsevytyĭ and G. M. Torbin, A certain class of JessenWintner type random variables, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki (1998), no. 4, 4854. (Ukrainian)
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 M. V. Pratsevytyĭ, Fractal Approach in Studies of Singular Distributions, Dragomanov National Pedagogical University Publishing House, Kyiv, 1998. (Ukrainian)
 19.
 G. M. Torbin, Multifractal analysis of singularly continuous probability measures, Ukrain. Mat. Zh. 57 (2005), no. 5, 837857; English transl. in Ukrainian Math. J. 57 (2005), no. 5, 837857. MR 2209816 (2007f:28010)
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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.unibonn.de
DOI:
http://dx.doi.org/10.1090/S009490002014009072
PII:
S 00949000(2014)009072
Keywords:
Infinite Bernoulli convolutions,
fractals,
singularly continuous probability measures,
HausdorffBesicovitch 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 1989/61, and Humboldt Foundation
Article copyright:
© Copyright 2014
American Mathematical Society
