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The distributions of random incomplete sums of a series with positive terms satisfying the property of non-linear homogeneity


Authors: M. V. Prats’ovytyĭ and I. O. Savchenko
Translated by: S. Kvasko
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 91 (2014).
Journal: Theor. Probability and Math. Statist. 91 (2015), 145-155
MSC (2010): Primary 60G30, 11K55, 28A80
DOI: https://doi.org/10.1090/tpms/973
Published electronically: February 4, 2016
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Abstract: The Lebesgue type as well as topological, metric, and fractal properties of the spectrum of the distribution of the random variable

$\displaystyle \xi =\sum _{n=1}^{\infty }a_n\xi _n $

are studied, where $ \sum _{n=1}^{\infty }a_n=a_1+a_2+\dots +a_n+r_n$ is a convergent series with positive terms such that $ r_{n+1}=a_{n+1}a_{n}$ for any $ n\in \mathbb{N}$ and $ (\xi _n)$ is a sequence of independent random variables taking only two values, 0 and $ 1$, with probabilities $ p_{0n}$ and $ p_{1n}$, respectively. We describe the point spectrum in the discrete case, and we prove that the distribution of $ \xi $ is of a Cantor singular type with an anomalous fractal spectrum in the continuous case. We also prove that the $ n$-fold convolution of the random variable $ \xi $ with itself has an anomalous fractal distribution.

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Additional Information

M. V. Prats’ovytyĭ
Affiliation: Department of Higher Mathematics, Institute for Physics and Mathematics, National Pedagogical Dragomanov University, Pirogov Street, 9, Kyiv 01130, Ukraine
Email: prats4@yandex.ru

I. O. Savchenko
Affiliation: Department of Fractal Analysis, Institute for Physics and Mathematics, National Pedagogical Dragomanov University, Pirogov Street, 9, Kyiv 01130, Ukraine
Email: igorsav4enko@ukr.net

DOI: https://doi.org/10.1090/tpms/973
Keywords: Bernoulli convolution, singularly continuous probability distribution, the set of incomplete sums of a series, Hausdorff--Besicovitch dimension of the spectrum of a probability distribution
Received by editor(s): June 14, 2014
Published electronically: February 4, 2016
Article copyright: © Copyright 2016 American Mathematical Society