A central limit theorem related to decimal and continued fraction expansion
type: Item
instance of: scholarly article
title: A central limit theorem related to decimal and continued fraction expansion
zbMATH Open document ID: 0921.11041
author: C. Faivre
published in: Archiv der Mathematik
publication date: 1999-09-29T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 11K50
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 11K55
zbMATH DE Number: 1186702
zbMATH Keywords: continued fraction expansion
zbMATH Keywords: strong mixing
zbMATH Keywords: central limit theorem
zbMATH Keywords: \(n\)-decimal approximations
zbMATH Keywords: ergodic theorem
reviewed by: Lothar Heinrich
MaRDI profile type: Publication
full work available at URL: s000130050219
OpenAlex ID: W2078996370
DOI: 10.1007/S000130050219
description: scientific article; zbMATH DE number 1186702