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.