Journal of the Mathematical Society of Japan

Berkes' limit theorem


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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

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

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Berkes' limit theorem Lacunary condition


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

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