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Probability Theory and Related Fields

, Volume 149, Issue 1–2, pp 139–148 | Cite as

A central limit theorem for trigonometric series with bounded gaps

  • Katusi Fukuyama
Article

Abstract

In this paper it is proved that there exists a sequence {n k } of integers with 1 ≤ n k+1n k ≤ 5 such that the distribution of \({(\cos 2\pi n_1 x + \dots + \cos 2\pi n_{N}) / \sqrt N}\) on ([ 0, 1 ], B, dx) converges to a Gaussian distribution. It gives an affirmative answer to the long standing problem on lacunary trigonometric series which ask the existence of series with bounded gaps satisfying a central limit theorem.

Keywords

Lacunary series The central limit theorem 

Mathematics Subject Classification (2000)

Primary 42A55 60F15 

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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of MathematicsKobe UniversityKobeJapan

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