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# Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia

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Volume 67, Issue 2

# On permutational invariance of the metric discrepancy results

Katusi Fukuyama
/ Yutaro Noda
Published Online: 2017-04-28 | DOI: https://doi.org/10.1515/ms-2016-0271

## Abstract

Let {nk} be a sequence of non-zero real numbers. We prove that the law of the iterated logarithm for discrepancies of the sequence {nkx} is permutational invariant if |nk+1/nk| → ∞ is satisfied.

MSC 2010: 11K38; 42A55; 60F15

Keywords: discrepancy; metric result; lacunary sequence

The first author is supported by KAKENHI 24340017 and 24340020.

## References

• [1]

Aistleitner, C.—Berkes, I.—Tichy, R.: Lacunary Sequences and Permutations. Dependence in Probability, Analysis and Number Theory, A volume in memory of Walter Philipp, (Eds. I. Berkes, R. Bradley, H. Dehling, M. Peligrad, and R. Tichy), Kendrick press, 2010, pp. 35–49.Google Scholar

• [2]

Aistleitner, C.—Berkes, I.—Tichy, R.: On the asymptotic behaviour of weakly lacunary series, Proc. Amer. Math. Soc. 139 (2011), 2505–2517.Google Scholar

• [3]

Aistleitner, C.—Berkes, I.—Tichy, R.: On the law of the iterated logarithm for permuted lacunary sequences, Proc. Steklov Inst. Math. 276 (2012), 3–20.

• [4]

Berkes, I.: On Strassen’s version of the log log law for multiplicative systems, Studia Sci. Math. Hungar. 8 (1973), 425–431.Google Scholar

• [5]

Dhompongsa, S.: Almost sure invariance principles for the empirical process of lacunary sequences, Acta Math. Hungar. 49 (1987), 83–102.Google Scholar

• [6]

Erdös, P.: Problems and results on diophantine approximations, Compos. Math. 16 (1964), 52–65.Google Scholar

• [7]

Fukuyama, K.: The law of the iterated logarithm for discrepancies of θn x, Acta Math. Hungar. 118 (2008), 155–170.Google Scholar

• [8]

Fukuyama, K.: The law of the iterated logarithm for the discrepancies of a permutation of nk x, Acta Math. Hungar. 123 (2009), 121–125.Google Scholar

• [9]

Fukuyama, K.: A Central Limit Theorem and a Metric Discrepancy Result for Sequence with Bounded Gaps. Dependence in Probability, Analysis and Number Theory, A volume in memory of Walter Philipp, (Eds. I. Berkes, R. Bradley, H. Dehling, M. Peligrad, and R. Tichy), Kendrick press, 2010, pp. 233–246.Google Scholar

• [10]

Fukuyama, K.: Metric discrepancy results for alternating geometric progressions, Monatsh. Math. 171 (2013), 33–63.

• [11]

Fukuyama, K.: A metric discrepancy result for the sequence of powers of minus two, Indag. Math. (NS) 25 (2014), 487–504.

• [12]

Fukuyama, K.—Mitsuhata, Y.: Bounded law of the iterated logarithm for discrepancies of permutations of lacunary sequences, Summer School on the Theory of Uniform Distribution, RIMS Kôkyûroku Bessatsu, B29 (2012), 65–88.Google Scholar

• [13]

Kuipers, L.—Niederreiter, H.: Uniform Distribution of Sequences, Wiley-Interscience, New York, 1974; Dover, New York, 2006.Google Scholar

• [14]

Philipp, W.: Limit theorems for lacunary series and uniform distribution mod 1, Acta Arith. 26 (1975), 241–251.Google Scholar

• [15]

Philipp, W.: A functional law of the iterated logarithm for empirical distribution functions of weakly dependent random variables, Ann. Probab. 5 (1977), 319–350.Google Scholar

• [16]

Takahashi, S.: An asymptotic property of a gap sequence, Proc. Japan Acad. 38 (1962), 101–104.Google Scholar

Accepted: 2015-05-27

Published Online: 2017-04-28

Published in Print: 2017-04-25

Citation Information: Mathematica Slovaca, Volume 67, Issue 2, Pages 349–354, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918,

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