Search Results
Abstract
This paper is mainly concerned with the limit distribution of 
on the unit interval when the increasing sequence {n
k} has bounded gaps, i.e., 1≤n
k+1−n
k=O(1). By Bobkov–Götze [4], it was proved that the limiting variance must be less than 1/2 in this case. They proved that the centered Gaussian distribution with variance 1/4 together with mixtures of Gaussian distributions belonging to a huge class can be limit distributions. In this paper it is proved that any Gaussian distribution with variance less than 1/2 can be a limit distribution.
Abstract
Let (nk)k≧1 be a lacunary sequence of positive integers, i.e. a sequence satisfying nk+1/nk > q > 1, k ≧ 1, and let f be a “nice” 1-periodic function with ∝01f(x) dx = 0. Then the probabilistic behavior of the system (f(nkx))k≧1 is very similar to the behavior of sequences of i.i.d. random variables. For example, Erdős and Gál proved in 1955 the following law of the iterated logarithm (LIL) for f(x) = cos 2πx and lacunary
Abstract
It is proved that two types of discrepancies of the sequence {θ n x} obey the law of the iterated logarithm with the same constant. The appearing constants are calculated explicitly for most of θ > 1.
5967 5982 Aistleitner, C. , Irregular discrepancy behavior of lacunary series, Monatsh. Math. , 160 (2010), no. 1, 1–29. MR 2610309 ( 2011i :11117
37 – 39 . [2] Gapoškin , V. F. 1966 Lacunary series and independent functions Uspehi Mat. Nauk 21 6
. Math. 25 241 251 GAPOSKIN, V. F., Lacunary series and independent functions, Uspehi Mat. Nauk 21 (1966
