On the stochastic behaviour of the digits in the modified Engel-type alternating series representations for real numbers (Q2770368)

Using a new algorithm of \textit{A. Knopfmacher} and \textit{J. Knopfmacher} [Int. J. Math. Math. Sci. 12, 603-613 (1989; Zbl 0683.10008)], the author shows that each real number \(x\) can be uniquely expressed in the form NEWLINE\[NEWLINEx= \alpha_0- \frac{1} {\alpha_1+1} \frac{1} {\alpha_2}+ \frac{1} {(\alpha_1+1)(\alpha_2+2)} \frac{1}{\alpha_3}-\cdots\;,NEWLINE\]NEWLINE where \(\alpha_n\geq 1\), \(\alpha_{n+1}\geq \alpha_n\) \((n=1,2,\dots)\). The digits \(\alpha_n= \alpha_n(x)\) are stochastically independent and identically distributed random variables with respect to the probability space \((I,{\mathcal B}_I,\lambda)\), where \(I= [0,1]\), \({\mathcal B}_I\) is the class of all Borel subsets of \(I\) and \(\lambda\) is the Lebesgue measure. Consequently classic limit theorems can be applied in the study of the metric properties of the sequence \((\alpha_n(x))_{n=1}^\infty\).NEWLINENEWLINEFor the entire collection see [Zbl 0969.00064].