Skip to main content
SpringerLink
Log in

Metric discrepancy results for subsequences of {θ k x}

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

We determine the set of possible constants appearing in the law of the iterated logarithm for discrepancies of subsequences of {θ k x}. We prove that the set coincides with the interval [ 1/2, Σ θ  ], where Σ θ is the constant for {θ k x}. It implies that the discrepancies decrease if we take a subsequence of {θ k x}.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aistleitner, C.: On the law of the iterated logarithm for the discrepancy of lacunary sequences. Trans. Am. Math. Soc. (to appear)

  2. Aistleitner, C., Berkes, I., Tichy, R.: Lacunary sequences and permutations. In: Dependence in Probability, Analysis and Number Theory. Walter Philipp Memorial Volume. Kendrick Press (to appear)

  3. Berkes, I.: On the asymptotic behaviour of ∑ f(n k x). I. Main Theorems, II. Applications. Z. Wahr. verv. Geb. 34, 319–345, 347–365 (1976)

  4. Csörgő M., Révész P.: Strong approximations in Probability and Statstics. Academic Press, New York (1981)

    Google Scholar 

  5. Fukuyama K.: The central limit theorem for Riesz–Raikov sums. Prob. Theory Relat. Fields 100, 57–75 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fukuyama K.: The law of the iterated logarithm for discrepancies of {θ n x}. Acta Math. Hungar 118, 155–170 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fukuyama K.: The law of the iterated logarithm for the discrepancies of a permutation of {n k x}. Acta Math. Hungar 123, 121–125 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fukuyama, K.: A central limit theorem and a metric discrepancy result for sequence with bounded gaps. In: Dependence in Probability, Analysis and Number Theory. Walter Philipp Memorial Volume. Kendrick Press (to appear)

  9. Kac M.: On the distribution of values of sums of type ∑ f(2k t). Ann. Math. 47, 33–49 (1946)

    Article  MATH  Google Scholar 

  10. Monrad W., Philipp D.: Nearby variables with nearby conditional laws and a strong approximation theorem for Hilbert space valued martingales. Probab. Theory Relat. Fields 88, 381–404 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  11. Philipp, W.: Almost sure invariance principles for partial sums of weakly dependent random variables. Memoirs Am. Math. Soc. 161 (1975)

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Strassen, V.: Almost sure behavior of sums of independent random variables and martingales. In: Fifth Berkeley Symposium on Mathematical Statistics and Probability, vol. 2, Part I, pp. 315–343 (1967)

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

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of mathematics, Kobe University, Rokko, Kobe, 657-8501, Japan

    Katusi Fukuyama

  2. Graduate School of Science, Kobe University, Rokko, Kobe, 657-8501, Japan

    Nobuhiko Hiroshima

Authors
  1. Katusi Fukuyama
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nobuhiko Hiroshima
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Katusi Fukuyama.

Additional information

The author was supported in part by KAKENHI 19204008.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fukuyama, K., Hiroshima, N. Metric discrepancy results for subsequences of {θ k x}. Monatsh Math 165, 199–215 (2012). https://doi.org/10.1007/s00605-010-0235-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00605-010-0235-7

Keywords

Mathematics Subject Classification (2000)

Search

Navigation