Acta Mathematica Hungarica

, Volume 129, Issue 4, pp 303–313 | Cite as

Pure Gaussian limit distributions of trigonometric series with bounded gaps

  • Katusi Fukuyama


This paper is mainly concerned with the limit distribution of \((\cos 2\pi n_{1}x+\cdots +\cos 2\pi n_{N}x)/\sqrt{N}\) on the unit interval when the increasing sequence {n k } has bounded gaps, i.e., 1≤n k+1n 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.

Key words and phrases

lacunary series 

2000 Mathematics Subject Classification

42A55 60F15 


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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2010

Authors and Affiliations

  1. 1.Department of MathematicsKobe UniversityKobeJapan

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