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On the almost sure central limit theorem and domains of attraction
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  • Published: 01 March 1995

On the almost sure central limit theorem and domains of attraction

  • István Berkes1 

Probability Theory and Related Fields volume 102, pages 1–17 (1995)Cite this article

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Summary

We give necessary and sufficient criteria for a sequence (X n) of i.i.d. r.v.'s to satisfy the a.s. central limit theorem, i.e.,

$$\mathop {\lim }\limits_{N \to \infty } \frac{1}{{\log N}}\sum\limits_{k \leqslant N} {\frac{1}{k}I} \left\{ {\frac{{S_k }}{{a_k }} - b_k< x} \right\} = \phi (x)\,\,\,\,{\text{a}}{\text{.s}}{\text{.}}\,\,{\text{for}}\,{\text{all}}\,x$$

for some numerical sequences (a n), (b n) whereS n=X 1+...+X n andI denotes indicator function. Our method leads also to new results on the limit distributional behavior ofS n/an−bn along subsequences (“partial attraction”), as well as to necessary and sufficient criteria for averaged versions of the central limit theorem such as

$$\mathop {\lim }\limits_{N \to \infty } \frac{1}{N}\sum\limits_{k \leqslant N} P \left( {\frac{{S_k }}{{a_k }} - b_k< x} \right) = \phi (x)\,\,\,\,\,{\text{for}}\,{\text{all}}\,x.$$

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References

  1. Berkes, I., Dehling, H.: Some limit theorems in log density. Ann. Probab.21, 1640–1670 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  2. Berkes, I., Dehling, H.: On the almost sure central limit theorem for random variables with infinite variance. J. Theor. Probab.7, 667–680 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  3. Berkes, I., Dehling, H., Móri, T.: Counterexamples related to the a.s. central limit theorem. Stud. Sci. Math. Hung.26, 153–164 (1991)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Feller, W.: An introduction to probability theory and its applications, Vol. II, 2nd Edition. New York: Wiley 1971

    MATH  Google Scholar 

  6. Fisher, A.: A pathwise central limit theorem for random walks (preprint 1989)

  7. Kuipers, L., Niederreiter, H.: Uniform distribution of sequences. New York: Wiley 1974

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Lévy, P.: Théorie de l'addition des variables aléatoires. Paris: Gauthier-Villars 1937

    MATH  Google Scholar 

  10. Loève, M.: Probability theory. New York: Van Nostrand (1955)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Mathematical Institute of the Hungarian Academy of Sciences, P.O.B. 127, H-1364, Budapest, Hungary

    István Berkes

Authors
  1. István Berkes
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Additional information

Research supported by Hungarian National Foundation for Scientific Research, Grant No. 1905

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Cite this article

Berkes, I. On the almost sure central limit theorem and domains of attraction. Probab. Theory Relat. Fields 102, 1–17 (1995). https://doi.org/10.1007/BF01295218

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  • Received: 28 February 1994

  • Revised: 04 October 1994

  • Published: 01 March 1995

  • Issue Date: March 1995

  • DOI: https://doi.org/10.1007/BF01295218

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Mathematics Subject Classification (1991)

  • 60F05
  • 60F15
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