# 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 (*n*_{k})_{k≧1} be a lacunary sequence of positive integers, i.e. a sequence satisfying *n*_{k+1}/*n*_{k} > *q* > 1, *k* ≧ 1, and let *f* be a “nice” 1-periodic function with ∝_{0}^{1}*f*(*x*) *dx* = 0. Then the probabilistic behavior of the system (*f*(*n*_{k}*x*))_{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 2*πx* and lacunary

*x*∈ (0, 1), where ‖

*f*‖

_{2}= (∝

_{0}

^{1}

*f*(

*x*)

^{2}

*dx*)

^{1/2}is the standard deviation of the random variables

*f*(

*n*

_{k}

*x*). If (

*n*

_{k})

_{k≧1}has certain number-theoretic properties (e.g.

*n*

_{k+1}/

*n*

_{k}→ ∞), a similar LIL holds for a large class of functions

*f*, and the constant on the right-hand side is always ‖

*f*‖

_{2}. For general lacunary (

*n*

_{k})

_{k≧1}this is not necessarily true: Erdős and Fortet constructed an example of a trigonometric polynomial

*f*and a lacunary sequence (

*n*

_{k})

_{k≧1}, such that the lim sup in the LIL (1) is not equal to ‖

*f*‖

_{2}and not even a constant a.e. In this paper we show that the class of possible functions on the right-hand side of (1) can be very large: we give an example of a trigonometric polynomial

*f*such that for any function

*g*(

*x*) with sufficiently small Fourier coefficients there exists a lacunary sequence (

*n*

_{k})

_{k≧1}such that (1) holds with √‖

*f*‖

_{2}

^{2}+

*g*(

*x*) instead of ‖

*f*‖

_{2}on the right-hand side.

## 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