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arXiv · 1311.4927

On the law of the iterated logarithm for permuted lacunary sequences

Christoph Aistleitner·Istvan Berkes·Robert Tichy
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Abstract

It is known that for any smooth periodic function $f$ the sequence $(f(2^kx))_{k\ge 1}$ 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 $(f(2^kx))_{k\ge 1}$ can ruin the validity of the law of the iterated logarithm, a very surprising result. In this paper we present an optimal condition on $(n_k)_{k\ge 1}$, 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 $(f(n_k x))_{k \geq 1}$. A similar result is proved for the discrepancy of the sequence $(\{n_k x\})_{k \geq 1}$, where $\{ \cdot \}$ denotes fractional part.

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Christoph Aistleitner, Istvan Berkes, Robert Tichy. 2014-01-10. On the law of the iterated logarithm for permuted lacunary sequences. https://arxiv.org/abs/1311.4927

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