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On the law of the iterated logarithm for trigonometric series with bounded gaps
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  • Published: 23 June 2011

On the law of the iterated logarithm for trigonometric series with bounded gaps

  • Christoph Aistleitner1 &
  • Katusi Fukuyama2 

Probability Theory and Related Fields volume 154, pages 607–620 (2012)Cite this article

  • 229 Accesses

  • 4 Citations

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Abstract

Let (n k ) k≥1 be an increasing sequence of positive integers. Bobkov and Götze proved that if the distribution of

$$\label{distr}\frac{\cos 2\pi n_1 x + \cdots +\cos 2 \pi n_N x}{\sqrt{N}}\qquad\quad\quad\quad (1)$$

converges to a Gaussian distribution, then the value of the variance is bounded from above by 1/2 − lim sup k/(2n k ). In particular it is impossible that for a sequence (n k ) k≥1 with bounded gaps (i.e. n k+1 − n k ≤ c for some constant c) the distribution of (1) converges to a Gaussian distribution with variance σ 2 = 1/2 or larger. In this paper we show that the situation is considerably different in the case of the law of the iterated logarithm. We prove the existence of an increasing sequence of positive integers satisfying

$$n_{k+1} - n_k \leq 2$$

such that

$$\limsup_{N \to \infty}\frac{\sum_{k=1}^N \cos 2 \pi n_k x}{\sqrt{2N \log \log N}} = +\infty \quad {a.e.}$$

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Author information

Authors and Affiliations

  1. Graz University of Technology, Institute of Mathematics A, Steyrergasse 30, Graz, 8010, Austria

    Christoph Aistleitner

  2. Department of Mathematics, Faculty of Science, Kobe University, Rokko, Kobe, 657-8501, Japan

    Katusi Fukuyama

Authors
  1. Christoph Aistleitner
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  2. Katusi Fukuyama
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Corresponding author

Correspondence to Christoph Aistleitner.

Additional information

Research supported by the Austrian Research Foundation (FWF), Project S9603-N23. Research supported in part by KAKENHI 19204008. This work was started during a stay of the authors in Oberwolfach in the framework of the “Oberwolfach Leibniz Fellows Programme”. The authors would like to thank the Mathematical Institute of Oberwolfach for its hospitality.

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Aistleitner, C., Fukuyama, K. On the law of the iterated logarithm for trigonometric series with bounded gaps. Probab. Theory Relat. Fields 154, 607–620 (2012). https://doi.org/10.1007/s00440-011-0378-z

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  • Received: 03 November 2010

  • Revised: 31 May 2011

  • Published: 23 June 2011

  • Issue Date: December 2012

  • DOI: https://doi.org/10.1007/s00440-011-0378-z

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Keywords

  • Law of the iterated logarithm
  • Bounded gaps

Mathematics Subject Classification (2010)

  • Primary 60F15
  • 11K38
  • 42A61
  • 42A32
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