Abstract
Let (n k ) k≥1 be an increasing sequence of positive integers. Bobkov and Götze proved that if the distribution of
converges to a Gaussian distribution, then the value of the variance is bounded from above by 1/2 − lim sup k/(2n k ). In particular it is impossible that for a sequence (n k ) k≥1 with bounded gaps (i.e. n k+1 − n k ≤ c for some constant c) the distribution of (1) converges to a Gaussian distribution with variance σ 2 = 1/2 or larger. In this paper we show that the situation is considerably different in the case of the law of the iterated logarithm. We prove the existence of an increasing sequence of positive integers satisfying
such that
Article PDF
Similar content being viewed by others
References
Aistleitner C.: Diophantine equations and the LIL for thediscrepancy of sub-lacunary sequences. Ill. J. Math. 53(3), 785–815 (2009)
Aistleitner C.: Irregular discrepancy behavior of lacunary series. Monatsh. Math. 160(1), 1–29 (2010)
Aistleitner C.: Irregular discrepancy behavior of lacunary series II. Monatsh. Math. 161(3), 255–270 (2010)
Aistleitner C.: On the law of the iterated logarithm for the discrepancy of lacunary sequences. Trans. Am. Math. Soc. 362(11), 5967–5982 (2010)
Berkes I.: A central limit theorem for trigonometric series with small gaps. Z. Wahrsch. Verw. Gebiete. 47(2), 157–161 (1979)
Berkes I., Philipp W.: The size of trigonometric and Walsh series and uniform distribution mod 1. J. Lond. Math. Soc. (2) 50(3), 454–464 (1994)
Berkes I., Philipp W., Tichy R.: Pair correlations and U-statistics for independent and weakly dependent random variables. Ill. J. Math. 45(2), 559–580 (2001)
Bobkov S.G., Götze F.: Concentration inequalities and limit theorems for randomized sums. Probab. Theory Relat. Fields 137(1-2), 49–81 (2007)
Drmota, M., Tichy, R.F.: Sequences, discrepancies and applications. In: Lecture Notes in Mathematics, vol. 1651. Springer, Berlin (1997)
Erdős, P., Gál, I.S.: On the law of the iterated logarithm. I, II. Nederl. Akad. Wetensch. Proc. Ser. A. 58(17), 65–76, 77–84 (1955)
Fukuyama K.: The law of the iterated logarithm for discrepancies of {θ n x}. Acta Math. Hung. 118(1–2), 155–170 (2008)
Fukuyama, K.: A central limit theorem and a metric discrepancy result for sequence with bounded gaps. In: Dependence in probability, analysis and number theory, a volume in memory of Walter Philipp, pp. 233–246. Kendrick press, Heber City (2010)
Fukuyama K.: A law of the iterated logarithm for discrepancies: non-constant limsup. Monatsh. Math. 160(2), 143–149 (2010)
Fukuyama K.: Pure gaussian limit distributions of trigonometric series with bounded gaps. Acta Math. Hung. 129, 303–313 (2010)
Fukuyama K.: A central limit theorem for trigonometric series with bounded gaps. Probab. Theory Related Fields. 149, 139–148 (2011)
Fukuyama, K., Miyamoto, S.: Metric discrepancy results for Erdős-Fortet sequence. Studia Sci. Math. Hung. (in press)
Fukuyama K., Nakata K.: A metric discrepancy result for the Hardy-Littlewood-Pólya sequences. Monatsh. Math. 160(1), 41–49 (2010)
Kesten, H.: The discrepancy of random sequences {kx}. Acta Arith. 10:183–213 (1964/1965)
Kuipers, L., Niederreiter, H.: Uniform distribution of sequences. In: Pure and applied Mathematics, Wiley-Interscience, Wiley, New York (1974)
Petrov V.V.: Limit theorems of probability theory Oxford studies in probability Sequences of independent random variables vol 4. The Clarendon Press Oxford University Press, New York (1995)
Philipp, W.: Limit theorems for lacunary series and uniform distribution mod 1. Acta Arith. 26(3):241–251 (1974/75)
Philipp W.: Empirical distribution functions and strong approximation theorems for dependent random variables. A problem of Baker in probabilistic number theory. Trans. Amer. Math. Soc. 345(2), 705–727 (1994)
Salem R., Zygmund A.: On lacunary trigonometric series. Proc. Nat. Acad. Sci. USA 33, 333–338 (1947)
Salem R., Zygmund A.: Some properties of trigonometric series whose terms have random signs. Acta Math. 91, 245–301 (1954)
Schoissengeier J.: A metrical result on the discrepancy of (n α). Glasgow Math. J. 40(3), 393–425 (1998)
Weiss M.: The law of the iterated logarithm for lacunary trigonometric series. Trans. Am. Math. Soc. 91, 444–469 (1959)
Author information
Authors and Affiliations
Corresponding author
Additional information
Research supported by the Austrian Research Foundation (FWF), Project S9603-N23. Research supported in part by KAKENHI 19204008. This work was started during a stay of the authors in Oberwolfach in the framework of the “Oberwolfach Leibniz Fellows Programme”. The authors would like to thank the Mathematical Institute of Oberwolfach for its hospitality.
Rights and permissions
About this article
Cite this article
Aistleitner, C., Fukuyama, K. On the law of the iterated logarithm for trigonometric series with bounded gaps. Probab. Theory Relat. Fields 154, 607–620 (2012). https://doi.org/10.1007/s00440-011-0378-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-011-0378-z