Journal of the Mathematical Society of Japan

Berkes' limit theorem

Satoshi TAKANOBU

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Abstract

In Berkes' striking paper of the early 1990s, he presented another limit theorem different from the central limit theorem for a lacunary trigonometric series not satisfying Erdős' lacunary condition. In this paper, we upgrade his result to the limit theorem having high versatility, which we would call Berkes' limit theorem. By this limit theorem, it is explained in a unified way that Fukuyama–Takahashi's counterexample and Takahashi's counterexample are all convergent to limiting distributions of the same type as Berkes.

Article information

Source
J. Math. Soc. Japan, Volume 71, Number 1 (2019), 117-145.

Dates
Received: 15 June 2017
Revised: 6 July 2017
First available in Project Euclid: 24 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1540368039

Digital Object Identifier
doi:10.2969/jmsj/78267826

Mathematical Reviews number (MathSciNet)
MR3909917

Zentralblatt MATH identifier
07056560

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 42A55: Lacunary series of trigonometric and other functions; Riesz products

Keywords
Berkes' limit theorem Lacunary condition

Citation

TAKANOBU, Satoshi. Berkes' limit theorem. J. Math. Soc. Japan 71 (2019), no. 1, 117--145. doi:10.2969/jmsj/78267826. https://projecteuclid.org/euclid.jmsj/1540368039


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References

  • [1] I. Berkes, Nongaussian limit distributions of lacunary trigonometric series, Can. J. Math., 43 (1991), 948–959.
  • [2] P. Erdős, On trigonometric sums with gaps, Magyar Tud. Akad. Mat. Kutato Int. Közl., 7 (1962), 37–42.
  • [3] K. Fukuyama and S. Takahashi, The central limit theorem for lacunary series, Proc. Amer. Math. Soc., 127 (1999), 599–608.
  • [4] S. Matsushita, Counterexample showing the best-possibility of Fukuyama–Takahashi's lacunary condition and limit theorem it satisfies, Master's thesis, Graduate School of Natural Science & Technology, Kanazawa University, 2009.
  • [5] S. Takahashi, On trigonometric series with gaps, Tôhoku Math. J., 17 (1965), 227–234.
  • [6] S. Takahashi, On lacunary trigonometric series, Proc. Japan Acad., 41 (1965), 503–506.
  • [7] S. Takahashi, On lacunary trigonometric series II, Proc. Japan Acad., 44 (1968), 766–770.