Faivre, C. On the central limit theorem for random variables related to the continued fraction expansion. (English) Zbl 0857.11038 Colloq. Math. 71, No. 1, 153-159 (1996). Applications of ergodic theory in the theory of continued fractions are based on the transformation \(T:[0,1] \to[0,1]\), \(T(0)=0\), \(T(x)= {1\over x} - [{1 \over x}]\) for \(x=[0;a_1(x), a_2(x), \dots] \in(0,1]\). It is well-known that \(([0,1],T,\nu)\) is an ergodic system, where \(\nu\) denotes the Gauss measure on \([0,1]\) defined by the density \(h(x)=(1/ \log 2) (1/(1+x))\).Unfortunately there are many important sequences of random variables that are not expressible in the form \(f(T^{n-1}(x))\) \((n=1,2, \dots)\) for a suitable \(f\). Such a sequence is \((\Theta_n (x))^\infty_{n=1}\), where \(|x-p_n/q_n |= \Theta_n(x)/q^2_n\) and \(p_n/q_n\) \((n=1,2, \dots)\) are convergents of \(x\). Define the mapping \[ W:[0,1]' \times [0,1] \to[0,1]' \times[0,1]; (x,y)\mapsto \left(T(x), {1\over a_1(x)+y} \right), \] where \([0,1]'\) denotes the set of all irrational numbers of \([0,1]\). Then \(\Theta_n(x)\) and also \(r_n(x)\) can be expressed in the form \(f(W^n(x),0)\) by a suitable function \(f\); here, \[ r_n(x) = \left|\left. x-{p_n \over q_n} \right|\right/ \left|x-{p_{n-1} \over q_{n-1}} \right|\qquad (n=1,2,\dots). \] The mapping \(W\) preserves the probability measure on \([0,1]' \times[0,1]\) defined by \[ d \mu (x,y) = {1\over\log 2} {dx dy \over(1+xy)^2} \] and \((W,\mu)\) is an ergodic system [cf. H. Nakada, Tokyo J. Math. 4, 399-426 (1981; Zbl 0479.10029)]. In the paper [W. Bosma, H. Jager and F. Wiedijk, Indag. Math. 45, 281-299 (1983; Zbl 0519.10043)] it is shown that \(\theta_n\) and \(r_n\) satisfy the strong law of large numbers.In this paper the central limit theorem for the random variables \(X_n(x) = f(W^n(x,t))\) is proved, where \(t\) is a fixed number in \([0,1]\). Reviewer: T.Šalát (Bratislava) Cited in 1 Document MSC: 11K50 Metric theory of continued fractions Keywords:continued fractions; sequences of random variables; ergodic system; central limit theorem Citations:Zbl 0479.10029; Zbl 0519.10043 PDFBibTeX XMLCite \textit{C. Faivre}, Colloq. Math. 71, No. 1, 153--159 (1996; Zbl 0857.11038) Full Text: DOI EuDML