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On the law of the iterated logarithm for permuted lacunary sequences

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Abstract

It is known that for any smooth periodic function f the sequence (f(2k x)) k≥1 behaves like a sequence of i.i.d. random variables; for example, it satisfies the central limit theorem and the law of the iterated logarithm. Recently Fukuyama showed that permuting (f(2k x)) k≥1 can ruin the validity of the law of the iterated logarithm, a very surprising result. In this paper we present an optimal condition on (n k ) k≥1, formulated in terms of the number of solutions of certain Diophantine equations, which ensures the validity of the law of the iterated logarithm for any permutation of the sequence (f(n k x)) k≥1. A similar result is proved for the discrepancy of the sequence ({n k x}) k≥1, where {·} denotes the fractional part.

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Authors and Affiliations

  1. Department of Analysis and Computational Number Theory, Graz University of Technology, Steyrergasse 30, 8010, Graz, Austria

    C. Aistleitner & R. Tichy

  2. Institute of Statistics, Graz University of Technology, Münzgrabenstraße 11, 8010, Graz, Austria

    I. Berkes

Authors
  1. C. Aistleitner
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  2. I. Berkes
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  3. R. Tichy
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Correspondence to C. Aistleitner.

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Dedicated to the memory of Professor Anatolii Alekseevich Karatsuba

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Aistleitner, C., Berkes, I. & Tichy, R. On the law of the iterated logarithm for permuted lacunary sequences. Proc. Steklov Inst. Math. 276, 3–20 (2012). https://doi.org/10.1134/S0081543812010026

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