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  • 1 University of Birmingham, Institute of Cancer and Genomic Sciences, Center for Computational Biology
  • | 2 Alfréd Rényi Institute of Mathematics, Reáltanoda u. 13{15, 1053 Budapest, Hungary
  • | 3 University of Keele, Research Institute for Primary Care and Health Sciences and Research Institute for Applied Clinical Sciences
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We prove that if Ik are disjoint blocks of positive integers and nk are independent random variables on some probability space (Ω,F,P) such that nk is uniformly distributed on Ik, then N1/2k=1N(sin2πnkxE(sin2πnkx)) has, with P-probability 1, a mixed Gaussian limit distribution relative to the probability space ((0, 1),B, λ), where B is the Borel σ-algebra and λ is the Lebesgue measure. We also investigate the case when nk have continuous uniform distribution on disjoint intervals Ik on the positive axis.

  • [1]

    Berkes, I., A central limit theorem for trigonometric series with small gaps, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 47 (1979), 157161.

    • Search Google Scholar
    • Export Citation
  • [2]

    Bobkov, S. and Götze, F., Concentration inequalities and limit theorems for randomized sums, Probab. Theory Related Fields, 137 (2007), 4981.

    • Search Google Scholar
    • Export Citation
  • [3]

    Erdős, P., On trigonometric sums with gaps, Magyar Tud. Akad. Mat. Kut. Int. Közl., 7 (1962), 3742.

  • [4]

    Fukuyama, K., A central limit theorem for trigonometric series with bounded gaps, Prob. Theory Rel. Fields, 149 (2011), 139148.

  • [5]

    Hartman, P.,. The divergence of non-harmonic gap series, Duke Math. J., 9 (1942), 404405.

  • [6]

    Kuipers, L. and Niederreiter, H., Uniform distribution of sequences, Wiley, 1974.

  • [7]

    Salem, R. and Zygmund, A., On lacunary trigonometric series, Proc. Nat. Acad. Sci. USA, 33 (1947), 333338.

  • [8]

    Salem, R. and Zygmund, A., Trigonometric series whose terms have random signs, Acta Math., 91 (1954), 245301.