Skip to main content

On Lacunary Series with Random Gaps

Abstract

Abstract We prove Strassen’s law of the iterated logarithm for sums \({\sum^{N}_{k=1} f(n_kx),}\) where f is a smooth periodic function on the real line and \({(n_k)_{k \geqq 1}}\) is an increasing random sequence. Our results show that classical results of the theory of lacunary series remain valid for sequences with random gaps, even in the nonharmonic case and if the Hadamard gap condition fails.

This is a preview of subscription content, access via your institution.

References

  1. Aistleitner C, Fukuyama K.: On the law of the iterated logarithm for trigonometric series with bounded gaps. Probab. Theory Related Fields, 154, 607–620 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  2. Berkes I.: On the central limit theorem for lacunary trigonometric series, Analysis Math.. , 4, 159–180 (1978)

    MathSciNet  MATH  Google Scholar 

  3. Berkes I.: A central limit theorem for trigonometric series with small gaps, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 47, 157–161 (1979)

    MathSciNet  Article  MATH  Google Scholar 

  4. I. Berkes and M. Weber, On the convergence of \({\sum c_kf(n_kx),}\) Mem. Amer. Math. Soc.,201 (2009), no. 943, viii+72 pp.

  5. Chover J.: On Strassen’s version of the loglog law, Z. Wahrsch. Verw. Gebiete, 8, 83–90 (1967)

    MathSciNet  Article  MATH  Google Scholar 

  6. Erdős P.: On trigonometric sums with gaps, Magyar Tud. Akad. Mat. Kut. Int. Közl., 7, 37–42 (1962)

    Google Scholar 

  7. Erdős P, Gál I. S.: On the law of the iterated logarithm. Proc. Nederl. Akad. Wetensch. Ser A, 58, 65–84 (1955)

    Google Scholar 

  8. K. Fukuyama, A central limit theorem and a metric discrepancy result for sequences with bounded gaps, in: Dependence in Probability, Analysis and Number Theory, Kendrick Press (2010), pp. 233–246.

  9. Fukuyama K.: A central limit theorem for trigonometric series with bounded gaps, Prob. Theory Rel. Fields, 149, 139–148 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  10. Gaposhkin V. F.: Lacunary series and independent functions, Russian Math. Surveys 21, 1–82 (1966)

    Article  MATH  Google Scholar 

  11. H. Halberstam and K. F. Roth, Sequences, Vol. I., Clarendon Press (Oxford, 1966).

  12. Kac M.: Probability methods in some problems of analysis and number theory, Bull. Amer. Math. Soc., 55, 641–665 (1949)

    Article  MATH  Google Scholar 

  13. Major P.: A note on Kolmogorov’s law of iterated logarithm, Studia Sci. Math. Hungar., 12, 161–167 (1977)

    MathSciNet  MATH  Google Scholar 

  14. Salem R, Zygmund A.: On lacunary trigonometric series, Proc. Nat. Acad. Sci. USA, 33, 333–338 (1947)

    MathSciNet  Article  MATH  Google Scholar 

  15. Schatte P.: On the asymptotic uniform distribution of sums reduced mod 1, Math. Nachr., 115(275–281), 115 275–281 (1984)

    MathSciNet  Google Scholar 

  16. Schatte P.: On a law of the iterated logarithm for sums mod 1 with applications to Benford’s law, Prob. Th. Rel. Fields, 77, 167–178 (1988)

    MathSciNet  Article  MATH  Google Scholar 

  17. Strassen V.: An invariance principle for the law of the iterated logarithm, Z. Wahrsch. Verw. Gebiete, 3(211–226), 3 211–226 (1964)

    MathSciNet  Google Scholar 

  18. Takahashi S.: On the law of the iterated logarithm for lacunary trigonometric series, Tohoku Math. J., 24, 319–329 (1972)

    MATH  Google Scholar 

  19. Weber M.: Discrepancy of randomly sampled sequences of integers, Math. Nachr., 271, 105–110 (2004)

    Article  MATH  Google Scholar 

  20. Weiss M.: On the law of the iterated logarithm for uniformly bounded orthonormal systems, Trans. Amer. Math. Soc., 92, 531–553 (1959)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to M. Raseta.

Additional information

Research supported by FWF Projekt W1230.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Raseta, M. On Lacunary Series with Random Gaps. Acta Math. Hungar. 144, 150–161 (2014). https://doi.org/10.1007/s10474-014-0430-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10474-014-0430-4

Key words and phrases

  • law of the iterated logarithm
  • lacunary series
  • random index

Mathematics Subject Classification

  • primary 60F17
  • 42A55
  • 42A61