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A central limit theorem for trigonometric series with bounded gaps

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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.

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Author information

Correspondence to Katusi Fukuyama.

Additional information

Dedicated to Professor Norio Kôno on his 70th birthday.

K. Fukuyama was supported in part by KAKENHI 17340029 and 19204008.

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Fukuyama, K. A central limit theorem for trigonometric series with bounded gaps. Probab. Theory Relat. Fields 149, 139–148 (2011).

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  • Lacunary series
  • The central limit theorem

Mathematics Subject Classification (2000)

  • Primary 42A55
  • 60F15