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.

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Published
29.04.2020
How to Cite
MorozM. 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