Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
On the central limit theorem for f (n k x)
Download PDF
Download PDF
  • Published: 16 December 2008

On the central limit theorem for f (n k x)

  • Christoph Aistleitner1 &
  • István Berkes2 

Probability Theory and Related Fields volume 146, Article number: 267 (2010) Cite this article

  • 213 Accesses

  • 29 Citations

  • Metrics details

Abstract

By a classical observation in analysis, lacunary subsequences of the trigonometric system behave like independent random variables: they satisfy the central limit theorem, the law of the iterated logarithm and several related probability limit theorems. For subsequences of the system ( f (nx)) n≥1 with 2π-periodic \({f\in L^2}\) this phenomenon is generally not valid and the asymptotic behavior of ( f (n k x)) k≥1 is determined by a complicated interplay between the analytic properties of f (e.g., the behavior of its Fourier coefficients) and the number theoretic properties of n k . By the classical theory, the central limit theorem holds for f (n k x) if n k  = 2k, or if n k+1/n k → α with a transcendental α, but it fails e.g., for n k  = 2k − 1. The purpose of our paper is to give a necessary and sufficient condition for f (n k x) to satisfy the central limit theorem. We will also study the critical CLT behavior of f (n k x), i.e., the question what happens when the arithmetic condition of the central limit theorem is weakened “infinitesimally”.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Aistleitner, C., Berkes, I.: On the law of the iterated logarithm for the discrepancy of \({\langle n_kx\rangle}\) . Monatshefte Math. (to appear)

  2. Baker R.C.: Metric number theory and the large sieve. J. Lond. Math. Soc. 24, 34–40 (1981)

    Article  MATH  Google Scholar 

  3. Berkes I.: On the asymptotic behaviour of ∑ f (n k x) I-II. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 34, 319–345 (1976) 347–365

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  6. Bobkov S., Götze F.: Concentration inequalities and limit theorems for randomized sums. Probab. Theory Relat. Fields 137, 49–81 (2007)

    Article  MATH  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  11. Fukuyama K., Petit B.: Le théorème limite central pour les suites de R. C. Baker. Ergodic Theory Dyn. Syst. 21, 479–492 (2001)

    MATH  MathSciNet  Google Scholar 

  12. Gaposhkin V.F.: Lacunary series and independent functions. Russian Math. Surv. 21, 3–82 (1966)

    Article  Google Scholar 

  13. Gaposhkin V.F.: The central limit theorem for some weakly dependent sequences. Theory Probab. Appl. 15, 649–666 (1970)

    Article  Google Scholar 

  14. Heyde C.C., Brown B.M.: On the departure from normality of a certain class of martingales. Ann. Math. Stat. 41, 2161–2165 (1970)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  17. Petit B.: Le théorème limite central pour des sommes de Riesz-Raikov. Probab. Theory Relat. Fields 93, 407–438 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  18. Philipp W.: Limit theorems for lacunary series and uniform distribution mod 1. Acta Arith. 26, 241–251 (1974–1975)

    MathSciNet  Google Scholar 

  19. Philipp, W., Stout, W.F.: Almost sure invariance principles for partial sums of weakly dependent random variables. Memoirs of the AMS No. 161 (1975)

  20. Philipp W.: Empirical distribution functions and strong approximation theorems for dependent random variables. A problem of Baker in probabilistic number theory. Trans. Am. Math. Soc. 345, 705–727 (1994)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  22. Salem R., Zygmund A.: La loi du logarithme itéré pour les séries trigonométriques lacunaires. Bull. Sci. Math. 74, 209–224 (1950)

    MATH  MathSciNet  Google Scholar 

  23. Takahashi S.: A gap sequence with gaps bigger than the Hadamards. Tôhoku Math. J. 13, 105–111 (1961)

    Article  MATH  Google Scholar 

  24. Takahashi S.: On lacunary trigonometric series. Proc. Jpn. Acad. 41, 503–506 (1965)

    Article  MATH  Google Scholar 

  25. Takahashi S.: On lacunary trigonometric series II. Proc. Jpn. Acad. 44, 766–770 (1968)

    Article  MATH  Google Scholar 

  26. Wintner A.: Diophantine approximation and Hilbert space. Am. J. Math. 64, 564–578 (1944)

    Article  MathSciNet  Google Scholar 

  27. Zygmund A.: Trigonometric series, 3rd edn, vol. I, II. Cambridge Mathematical Library. Cambridge University Press, London (2002)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics A, Graz University of Technology, Steyrergasse 30, 8010, Graz, Austria

    Christoph Aistleitner

  2. Institute of Statistics, Graz University of Technology, Münzgrabenstrasse 11, 8010, Graz, Austria

    István Berkes

Authors
  1. Christoph Aistleitner
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. István Berkes
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christoph Aistleitner.

Additional information

C. Aistleitner was research supported by FWF grant S9603-N13.

I. Berkes was research supported by FWF grant S9603-N13 and OTKA grants K 61052 and K 67961.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Aistleitner, C., Berkes, I. On the central limit theorem for f (n k x). Probab. Theory Relat. Fields 146, 267 (2010). https://doi.org/10.1007/s00440-008-0190-6

Download citation

  • Received: 29 May 2008

  • Revised: 23 September 2008

  • Published: 16 December 2008

  • DOI: https://doi.org/10.1007/s00440-008-0190-6

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Lacunary series
  • Central limit theorem
  • Diophantine equations

Mathematics Subject Classification (2000)

  • Primary 42A55
  • 60F05
  • 11D04
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature