- Autor
- Bazarova, Alina
- Berkes, István
- Horváth, Lajos
- TitelOn the Extremal Theory of Continued Fractions
- Datei
- DOI10.1007/s10959-014-0577-5
- Erschienen inJournal of Theoretical Probability
- Band29
- Erscheinungsjahr2016
- Heft1
- Seiten248-266
- LicenceCC BY
- ISSN1572-9230
- Download Statistik2051
- Peer ReviewJa
- AbstractLetting \(x = [a 1 (x), a 2 (x), . . .]\) denote the continued fraction expansion of an irrational 1 number \(x ∈ (0, 1)\), Khinchin proved that \(S_n (x) = Σ^n_{k=1} a_k (x) ∼ \frac{1}{log 2} n log n\) in measure, but not for almost every x. Diamond and Vaaler showed that removing the largest term from \(S_n (x)\), the pre- vious asymptotics will hold almost everywhere, showing the crucial influence of the extreme terms of \(S_n (x)\) on the sum. In this paper we determine, for \(d_n → ∞\) and \(d_n/n → 0\), the precise asymptotics of the sum of the \(d_n\) largest terms of \(S_n (x)\) and show that the sum of the remaining terms has an asymptotically Gaussian distribution.
[136.34 KB]