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- Title
Singularity and Fine Fractal Properties of One Class of Infinite Bernoulli Convolutions with Essential Overlapping. II.
- Authors
Lebid', M.; Torbin, H.
- Abstract
We study the Lebesgue structure and fine fractal properties of infinite Bernoulli convolutions, i.e., the distributions of random variables of the form ξ = ∑ ξ a, where ∑ a is a convergent positive series and ξ are independent (generally speaking, nonidentically distributed) Bernoulli random variables. Our main aim is to investigate the class of Bernoulli convolutions with essential overlaps generated by a series ∑ a such that, for any k ∈ ℕ, there exists s ∈ ℕ ⋃{0} for which a = a =...= $$ {a}_{k+{s}_k} $$ ≥ $$ {r}_{k+{s}_k} $$ and, in addition, s > 0 for infinitely many indices k. In this case, almost all (both in a sense of Lebesgue measure and in a sense of fractal dimension) points of the spectrum have continuum many representations of the form ξ = ∑ ε a with ε ∈ {0, 1}. It is shown that the probability measure μ has either a pure discrete distribution or a pure singularly continuous distribution. We also establish sufficient conditions for the confidentiality of the family of cylindrical intervals on the spectrum $$ {S}_{\mu_{\xi }} $$ generated by the distributions of the random variable ξ. In the case of singularity, we also deduce the explicit formula for the Hausdorff dimension of the corresponding probability measure [i.e., the Hausdorff-Besicovitch dimension of the minimal supports of the measure μ (in a sense of dimension)].
- Subjects
BERNOULLI equation; MATHEMATICAL convolutions; RANDOM variables; DISCRETE uniform distribution; MATHEMATICAL analysis
- Publication
Ukrainian Mathematical Journal, 2016, Vol 67, Issue 12, p1884
- ISSN
0041-5995
- Publication type
Article
- DOI
10.1007/s11253-016-1197-2