Skip to main content
SpringerLink
Log in

Critical Behavior in Almost Sure Central Limit Theory

  • Published:
Journal of Theoretical Probability Aims and scope Submit manuscript

Abstract

Let X 1,X 2,… be i.i.d. random variables with EX 1=0, EX 21 =1 and let S k =X 1+⋅⋅⋅+X k . We study the a.s. convergence of the weighted averages

$$D_{N}^{-1}\sum_{k=1}^{N}d_{k}I\biggl\{\frac{S_{k}}{\sqrt{k}}\leq x\biggr\},$$

where (d k ) is a positive sequence with D N =∑ N k=1 d k →∞. By the a.s. central limit theorem, the above averages converge a.s. to Φ(x) if d k =1/k (logarithmic averages) but diverge if d k =1 (ordinary averages). Under regularity conditions, we give a fairly complete solution of the problem for what sequences (d k ) the weighted averages above converge, resp. the corresponding LIL and CLT hold. Our results show that logarithmic averaging, despite its prominent role in a.s. central limit theory, is far from optimal and considerably stronger results can be obtained using summation methods near ordinary (Cesàro) summation.

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. Atlagh, M., Weber, M.: Le théorème central limite presque sûr. Expo. Math. 18, 97–126 (2000)

    MATH  MathSciNet  Google Scholar 

  2. Becker-Kern, P.: Almost sure limit theorems of mantissa type for semistable domains of attraction. Acta Math. Hung. 114, 301–336 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  3. Berkes, I.: Results and problems related to the pointwise central limit theorem. In: Asymptotic Methods in Probability and Statistics (Ottawa, ON, 1997), pp. 59–96. North-Holland, Amsterdam (1998)

    Google Scholar 

  4. Berkes, I., Csáki, E.: A universal result in almost sure central limit theory. Stoch. Process. Appl. 94, 105–134 (2001)

    Article  MATH  Google Scholar 

  5. Berkes, I., Horváth, L.: Almost sure invariance principles for logarithmic averages. Studia Sci. Math. Hung. 33, 1–24 (1997)

    MATH  Google Scholar 

  6. Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Encyclopedia of Mathematics and its Applications, vol. 27. Cambridge (1987)

  7. Brosamler, G.A.: An almost everywhere central limit theorem. Math. Proc. Camb. Philos. Soc. 104, 561–574 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  8. Chandrasekharan, K., Minakshisundaram, S.: Typical Means. Oxford University Press (1952)

  9. Fisher, A.: A pathwise central limit theorem for random walks. Preprint (1989)

  10. Hörmann, S.: An extension of almost sure central limit theory. Stat. Probab. Lett. 76, 191–202 (2006)

    MATH  Google Scholar 

  11. Lacey, M.T., Philipp, W.: A note on the almost sure central limit theorem. Stat. Probab. Lett. 9, 201–205 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  12. Lévy, P.: Théorie de l’addition des variables alétoires, 12th edn. Jacques Gabay (2003)

  13. Major, P.: An improvement of Strassen’s invariance principle. Ann. Probab. 7, 55–61 (1979)

    MATH  MathSciNet  Google Scholar 

  14. Peligrad, M., Révész, P.: On the almost sure central limit theorem. In: Almost Everywhere Convergence, II (Evanston, IL, 1989), pp. 209–225. Academic, Boston (1991)

    Google Scholar 

  15. Petrov, V.: Limit Theorems of Probability Theory. Oxford Science Publications, Oxford (1995)

    MATH  Google Scholar 

  16. Rozanov, Y.A.: Stationary Random Processes. Holden-Day, San Francisco (1967)

    MATH  Google Scholar 

  17. Schatte, P.: On strong versions of the central limit theorem. Math. Nachr. 137, 249–256 (1988)

    Article  MATH  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  MathSciNet  Google Scholar 

  19. Weber, M.: Un théorème central limite presque sûr à moments généralisés pour les rotations irrationnelles. Manuscripta Math. 101, 175–190 (2000)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Statistics, Graz University of Technology, Steyrergasse 17/IV, 8010, Graz, Austria

    Siegfried Hörmann

Authors
  1. Siegfried Hörmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Siegfried Hörmann.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hörmann, S. Critical Behavior in Almost Sure Central Limit Theory. J Theor Probab 20, 613–636 (2007). https://doi.org/10.1007/s10959-007-0080-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10959-007-0080-3

Keywords

Search

Navigation