Numerical characteristics of the random variable associated with the expansions of real numbers by the Engel series

Abstract

It is known that any $x \in \left(0;1\right]\equiv\Omega$ has a unique Engel expansion $$ \displaystyle x=\sum\limits_{n=1}^{\infty}\frac{1}{\left(p_1(x)+1\right)\ldots \left(p_n(x)+1\right)},$$ where $p_n(x)\in\mathbb{N},$ $p_{n+1}(x)\geq p_n(x)$ for all $n \in \mathbb{N}.$ This means that $p_n(x)$ is a well-defined measurable function on the probability space $ (\Omega, \mathcal {F}, \lambda),$ where $\mathcal{F}$ is the $\sigma$-algebra of Lebesgue-measurable subsets of $\Omega$ and $\lambda$ is the Lebesgue measure.

 

The main subject of our research is the function $$\psi(x)=\sum_{n=1}^{\infty}\frac{1}{p_n(x)+1},$$ defined on $\Omega^*\subset\Omega,$ where $ \Omega^* $ is the convergence set of the series $\displaystyle\sum\nolimits_{n=1}^{\infty}\dfrac{1}{p_n(x)+1}.$ We prove that the function $\psi$ is defined a.e. on $(0;1]$ and $\psi$ is a random variable on the probability space $(\Omega^*, \mathcal{F^*}, \lambda),$ where $\mathcal{F^*}$ is the $\sigma$-algebra of Lebesgue-measurable subsets of $\Omega^*,$ and obtain the mathematical expectation and variance of the function $\psi.$ Also, we consider the  variables $\psi_k$ as a generalization of the function $ \psi $ and calculate the mathematical expectations $M\psi_k $ of these random variables.

References

Pratsiovytyi, Mykola; Khvorostina, Yuriy. Topological and metric properties of distributions of random variables represented by the alternating Lüroth series with independent elements. Random Oper. Stoch. Equ. 21 (2013), no. 4, 385--401. https://doi.org/10.1515/rose-2013-0018 DOI: https://doi.org/10.1515/rose-2013-0018

Shallit, J. O. Metric theory of Pierce expansions. Fibonacci Quart. 24 (1986), no. 1, 22--40. https://www.fq.math.ca/Scanned/24-1/shallit.pdf

Zhykharyeva, Yulia; Pratsiovytyi, Mykola. Expansions of numbers in positive Lüroth series and their applications to metric, probabilistic and fractal theories of numbers. Algebra Discrete Math. 14 (2012), no. 1, 145--160. http://admjournal.luguniv.edu.ua/index.php/adm/article/view/716/248

Барановський, О. М.; Працьовитий, М. В.; Торбiн, Г. М. Ряди Остроградського–Серпiнського–Пiрса та їхнi застосування, Наук. думка, Київ (2013) [Baranovs'kyy̆, O. M.; Prac'ovytyy̆, M. V.; Torbin, G. M. Rjady Ostrograds'kogo–Serpins'kogo–Pirsa ta ïhni zastosuvannja, Nauk. dumka, Kyïv (2013)]. https://scholar.google.com/citations?user=1V2cuyQAAAAJ&hl=en&oi=sra

Kolmogorov, A. N.; Fomin, S. V. Элементы теории функций и функционального анализа. (Russian) [Elements of the theory of functions and functional analysis] Fourth edition, revised. Izdat. "Nauka", Moscow, 1976. 543 pp. http://fulviofrisone.com/attachments/article/485/Kolmogorov-Fomin.pdf

Працьовита, I. М.; Заднiпряний, М. В. Розклади чисел в ряди Сiльвестера та їх застосування. Наук. часопис Нац. пед. ун-ту iм. М. П. Драгоманова. Фiз.-мат. науки, no. 10, 73--87 (2009) [Prac'ovyta, I. M.; Zadniprjanyy̆, M. V. Rozklady chysel v rjady Sil'vestera ta ïh zastosuvannja. Nauk. chasopys Nac. ped. un-tu im. M. P. Dragomanova. Fiz.-mat. nauky, no. 10, 73--87 (2009).].

Працьовитий, М. В.; Гетьман, Б. I. Ряди Енгеля та їх застосування. Наук. часопис Нац. пед. ун-ту iм. М. П. Драгоманова. Фiз.-мат. науки, no. 7, 105--116 (2006) [Prac'ovytyy̆, M. V.; Get'man, B. I. Rjady Engelja ta ïh zastosuvannja. Nauk. chasopys Nac. ped. un-tu im. M. P. Dragomanova. Fiz.-mat. nauky, no. 7, 105--116 (2006)].

Смирнов, В. И. Курс высшей математики, т. 5. Наука, Москва (1959) [Smirnov, V. I. Kurs vyssheĭ matematiki, t. 5. Nauka, Moskva (1959)].

Published
29.04.2020
How to Cite
Moroz, M. P. “Numerical Characteristics of the Random Variable Associated With the Expansions of Real Numbers by the Engel Series”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, no. 5, Apr. 2020, pp. 658–666, doi:10.37863/umzh.v72i5.2284.
Section
Research articles