A central limit theorem related to decimal and continued fraction expansion
The main result of this paper is a central limit theorem for the exact number \(k_n(x)\) of partial quotients of \(x\in (0,1)\) which can be obtained from \(d_n(x)= 10^{-n} [10^n]\) and \(e_n(x)= d_n(x)+ 10^{-n}\), the two consecutive \(n\)-decimal approximations of \(x\). It could be shown that \[ \lim_{n\to\infty} m\{x\in (0,1): (k_n(x)- an)/\sigma\sqrt{n}\leq z\}= \Phi(z) \] for some constant \(\sigma>0\), where \(a= 6\log 2\log 10/\pi^2\), \(m(\cdot)\) is the Lebesgue measure and \(\Phi(\cdot)\) is the standard normal distribution function. This result is an extension of a (in some sense surprising) ergodic theorem \(k_n(x)/n\to a\) as \(n\to\infty\) for a.e. \(x\in (0,1)\) which has been proved by \textit{G. Lochs} [Abh. Math. Semin. Univ. Hamb. 27, 142-144 (1964; Zbl 0124.28003)].
scientific article; zbMATH DE number 1186702