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
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
MathSciNet review:
3364130
Full-text PDF Free Access
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Additional Information
Abstract: The Lebesgue type as well as topological, metric, and fractal properties of the spectrum of the distribution of the random variable \[ \xi =\sum _{n=1}^{\infty }a_n\xi _n \] 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.
References
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References
- S. Albeverio, O. Baranovskyi, M. Prats’ovytyĭ, and G. Torbin, The set of incomplete sums of the first Ostrogradsky series and anomalously fractal probability distributions on it, Rev. Roum. Math. Pures. Appl. 54 (2009), no. 2, 85–115. MR 2519524 (2011a:60054)
- S. Albeverio and G. Torbin, On fine fractal properties of generalized infinite Bernoulli convolutions, Bull. Sci. Math. 132 (2008), no. 8, 711–727. MR 2474489 (2010e:28006)
- B. Jessen and A. Wintner, Distribution function and the Riemann Zeta-function, Trans. Amer. Math. Soc. 38 (1935), no. 1, 48–88. MR 1501802
- S. Kakeya, On the partial sums of an infinite series, Tôhoku Sci. Rep. 3 (1915), no. 4, 159–164.
- P. Lévy, Sur les séries dont les termes sont des variables éventuelles indépendantes, Studia Math. 3 (1931), 119–155.
- Y. Peres, W. Schlag, and B. Solomyak, Sixty year of Bernoulli convolutions, Fractal Geometry and Stochastics II. Progress in Probability 46 (2000), 39–65. MR 1785620 (2001m:42020)
- M. V. Prats’ovytyĭ and O. Y. Feshchenko, Topological, metric and fractal properties of probability distributions on the set of incomplete sums of positive series, Theory Stoch. Process. 13(29) (2007), no. 1–2, 205–224. MR 2343824 (2009a:28025)
- B. Solomyak, On the random series $\sum \pm \lambda ^{n}$ (an Erdős problem), Ann. Math. 142 (1995), 611–625. MR 1356783 (97d:11125)
- O. M. Baranovs’kyĭ, M. V. Pratsyovytyĭ, and G. M. Torbin, Ostrograds’kyĭ–Sierpiński–Pierce Series and their Applications, “Naukove dumka”, Kyiv, 2013. (Ukrainian)
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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
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