Числові характеристики випадкової величини, пов'язаної з представленням дійсних чисел рядами Остроградського-Серпінського-Пірса

Abstract

It is known that any irrational number $x\in\left(0;1\right)\backslash \mathbb{Q}\equiv\Omega$ has a unique Ostrogradsky-Sierpinski-Pierce expansion: $$x=\sum_{n=1}^{\infty}\frac{1}{q_1(x)\cdot...\cdot q_n(x)},$$ where $q_n(x)\in\mathbb{N}$, $q_{n+1}(x)> q_n(x)$, for all $n \in \mathbb{N}$.
To represent an irrational number $x\in\Omega$ by Ostrogradsky-Serpinsky-Pierce expansion we have calculated numerical characteristics of the random variable $$\xi(X)=\sum_{n=1}^{\infty}\frac{1}{q_n(X)},$$ where $X$ is uniform distribution on $\Omega$. A new method for calculating the mathematical expectation is proposed, which differs from the method described in \cite{Shallit1986}, and we have calculated variance $D\xi$. We consider the random variables $\xi_n$ as a
generalization of the function $\xi$ and we have calculated mathematical expectations $M\xi_n$ of them.