On the sum of the partial quotients of the continued fraction expansion (Q2759159)

Let \([a_1(x), a_2(x),\dots]\) be the regular continued fraction expansion of a real number \(x\in (0,1)\). The authors prove some strong LLN-type results for truncated partial quotients \((a_i)_{i\geq 1}\) which form an exponentially \(\psi\)-mixing sequence of random variables with infinite mean. The main result is the following limit theorem: For (Lebesgue-) almost all \(x\in (0,1)\) NEWLINE\[NEWLINE\lim_{n\to\infty} \frac{1}{n\log n} \sum_{i=1}^n a_i(x) \mathbf{1}_{(0,n^\alpha (\log n)^\beta]} (a_i(x))= \frac{\alpha}{\log 2},NEWLINE\]NEWLINE where either \(0< \alpha< 1\), \(\beta\geq 0\) or \(\alpha=1\), \(0\leq \beta<1\).