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  • Author or Editor: Katusi Fukuyama x
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Summary  

We prove that the gap seriesΣf(n k x) does not behave like an independent random series when f is a function of bounded variation with rational discontinuity.

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Abstract  

Takahashi [15] gave a concrete upper bound estimate of the law of the iterated logarithm for ∑ f(n k x). We extend this result and prove the best possibility of this bound.

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Abstract

This paper is mainly concerned with the limit distribution of on the unit interval when the increasing sequence {n k} has bounded gaps, i.e., 1≤n k+1n k=O(1). By Bobkov–Götze [4], it was proved that the limiting variance must be less than 1/2 in this case. They proved that the centered Gaussian distribution with variance 1/4 together with mixtures of Gaussian distributions belonging to a huge class can be limit distributions. In this paper it is proved that any Gaussian distribution with variance less than 1/2 can be a limit distribution.

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For the classical Erdős-Fortet sequence n k = 2k − 1 we show that the law of the iterated logarithm for star discrepancy of {n k x} has non-constant limsup, while the law for discrepancy has constant limsup.

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