Central limit theorems for the ergodic adding machine
type: Item
instance of: scholarly article
title: Central limit theorems for the ergodic adding machine
zbMATH Open document ID: 1023.60028
DOI: 10.1007/BF02787403
author: Mordechay B. Levin
author: Ely Merzbach
published in: Israel Journal of Mathematics
publication date: 2003-07-27T00:00:00Z
review text: In several papers \textit{J. Beck} [see, e.g., in: Sets, graphs and numbers. Colloq. Soc. János Bolyai 60, 23-66 (1992; Zbl 0783.60013)] found some results on the stochastic behavior of the rotation on the circle. According to Roth's theorem, there is a first class of sequences \((\beta_n)_{n=0}^{\infty}\) of random numbers from the unit interval such that the triangular array with random variable \({\mathbf 1}_{[u,u+y)}(\beta_n)-y\) satisfies the CLT with extremely small standard deviation. In the present paper, the authors find a second class of such sequences of random numbers \((x_n)_{n=1}^{\infty}\) (the orbit of the ergodic adding machine) such that the corresponding sequences of zeros and ones \({\mathbf 1}_{[0,y)}(x_n)\) satisfy CLT with extremely small standard deviation \(\sigma_N=O(\sqrt{\log N}),\) instead of \(O(\sqrt{N}),\) as \(N\to +\infty\) (as in Roth's theorem). The detailed description of the ergodic adding machine is given, for example, by \textit{N. A. Friedman} [``Introduction to ergodic theory'' (1970; Zbl 0212.40004)].
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 37A50
zbMATH DE Number: 1932067
zbMATH Keywords: random numbers
zbMATH Keywords: ergodic adding machine
zbMATH Keywords: orbits
zbMATH Keywords: central limit theorem
zbMATH Keywords: standard deviation
reviewed by: Anatoliy Swishchuk
MaRDI profile type: Publication
cites work: Q4226441
cites work: Q5614084
cites work: Q3962219
cites work: Q4767350
cites work: Van der Corput sequences, Kakutani transforms and one-dimensional numerical integration
description: scientific article; zbMATH DE number 1932067