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Theory of Probability and Mathematical Statistics

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Properties and distributions of values of fractal functions related to $ \boldsymbol{Q_2}$-representations of real numbers


Authors: M. V. Pratsiovytyi and S. P. Ratushniak
Translated by: N. N. Semenov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 99 (2018).
Journal: Theor. Probability and Math. Statist. 99 (2019), 211-228
MSC (2010): Primary 60E05, 28A80, 97F50, 26A30
DOI: https://doi.org/10.1090/tpms/1091
Published electronically: February 27, 2020
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Abstract | References | Similar Articles | Additional Information

Abstract: A $ Q_2$-representation of numbers $ x\in [0;1]$ is determined by a parameter $ q_0\in (0;1)$ and provides an expansion of a number $ x\in [0;1]$ in the form of the following series:

$\displaystyle x=\alpha _1q_{1-\alpha _1}+\sum _{k=2}^{\infty }{\left (\alpha _k... ...x)}}\right )}\equiv \Delta ^{Q_2}_{\alpha _1\alpha _2\ldots \alpha _n\ldots }, $

where $ \alpha _k\in \{0,1\}\equiv A$ and $ q_1\equiv 1-q_0$. The structural and local as well as global topological/metric and fractal properties are studied for the function $ f_{\varphi }$ defined by

$\displaystyle f_{\varphi }(x)$ $\displaystyle =f_{\varphi }\left (\Delta ^{Q_2}_{\alpha _1\alpha _2\alpha _3\ldots \alpha _{n-1}\alpha _{n}\alpha _{n+1}\ldots }\right )$    
  $\displaystyle =\Delta ^{Q_2}_{\varphi (\alpha _1,\alpha _2)\varphi (\alpha _2,\... ...\varphi (\alpha _{n-1},\alpha _{n})\varphi (\alpha _{n},\alpha _{n+1})\ldots },$    

where $ \varphi $ is a given function, $ \varphi \colon A^2\rightarrow A$.

For the random variable $ Y=f_{\varphi }(X)$ where $ X$ is a random variable with a known distribution, we study its Lebesgue structure (the discrete, absolutely continuous, and singular components) and spectral properties of the set of points of growth of the distribution function of $ Y$.


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Additional Information

M. V. Pratsiovytyi
Affiliation: Dragomanov National Pedagogical University, Pyrogova Street, 9, 01601, Kyiv, Ukraine
Email: prats4444@gmail.com

S. P. Ratushniak
Affiliation: Institute of Mathematics, National Academy of Science of Ukraine, Tereshchenkivs’ka Street, 3, 01004, Kyiv, Ukraine
Email: ratush404@gmail.com

DOI: https://doi.org/10.1090/tpms/1091
Keywords: $Q_2$-representations of the fractional part of a real number, classical binary representation of real numbers, operator of the left shift of digits of the representation, inversor of digits of the representation, singular function, fractal function, the level set of a function, distribution of values of a function
Received by editor(s): August 16, 2018
Published electronically: February 27, 2020
Article copyright: © Copyright 2020 American Mathematical Society