Central limit theorems for the ergodic adding machine
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)].
scientific article; zbMATH DE number 1932067