Skip to main content
SpringerLink
Log in

Central limit theorems for the ergodic adding machine

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this paper we find a second class of sequences of random numbers (x n ) ∞ n=1 (the orbit of the ergodic adding machine) such that the corresponding sequences of zeros and ones 1[0,y](x n) (n=1,2,...,N) satisfy Central Limit Theorems with extremely small standard deviationσ N=O(√logN), instead ofO(√N), asN → ∞.

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.

Similar content being viewed by others

References

  1. J. Beck,Randomness of n√2mod 1 and a Ramsey property of the hyperbola, inSets, Graphs and Numbers (Budapest, 1991), Colloquia Mathematica Societatis János Bolyai, 60 (G. Halász, L. Lovász, D. Miklós and I. Szöny, eds.), North-Holland, Amsterdam, 1992, pp. 23–66.

    Google Scholar 

  2. J. Beck,Diophantine approximation and quadratic fields, inNumber Theory (Egez, 1996), (K. Györy, A. Pethö and V. Sós, eds.), de Gruyter, Berlin, 1998, pp. 55–93.

    Google Scholar 

  3. J. Beck,From probabilistic Diophantine approximation to quadratic fields, inRandom and Quasi-Random Points Sets (P. Kellekalek and G. Larcher, eds.), Lecture Notes in Statistics, Vol. 138, Springer, New York, 1998, pp. 1–48.

    Google Scholar 

  4. S. N. Bernstein,Sur l’extension du théorème limite du calcul des probabilités aux sommes de quantités dépendantes, Mathematische Annalen97 (1926), 1–59.

    Article  Google Scholar 

  5. N. A. Friedman,Introduction to Ergodic Theory, Van Nostrand Reinhold Co., New York, 1970.

    MATH  Google Scholar 

  6. P. Hall,Rates of Convergence in the Central Limit Theorem, Pitman Advanced Publishing Program, Boston, 1982.

    MATH  Google Scholar 

  7. L. Kuipers and H. Niederreiter,Uniform Distribution of Sequences, Wiley, New York, 1974.

    MATH  Google Scholar 

  8. B. Lapeyre and G. Pages,Families of sequences with low discrepancy obtained by iteration of transforms of [0,1], Comptes Rendus de l’Académie des Sciences, Paris, Série I308 (1989), 507–509.

    MATH  MathSciNet  Google Scholar 

  9. M. Levin,Stochastic behavior of the ergodic adding machine, inProceedings of the Annual Meeting of the Israel Mathematical Union, Haifa University, Haifa, 2000, p. 41.

    Google Scholar 

  10. G. Pages,Van der Corput sequences, Kakutani transforms and one-dimensional numerical integration, Journal of Computational and Applied Mathematics44 (1992), 21–39.

    Article  MATH  MathSciNet  Google Scholar 

  11. K. Petersen,Ergodic Theory, Cambridge University Press, 1983.

  12. P. D. Proinov and E. V. Atanasov,On the distribution of the van der Corput generalized sequences, Comptes Rendus de l’Académie des Sciences, Paris, SérieI 307 (1988), 895–900.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Bar-Ilan University, 52900, Ramat-Gan, Israel

    Mordechay B. Levin & Ely Merzbach

Authors
  1. Mordechay B. Levin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ely Merzbach
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mordechay B. Levin.

Additional information

Dedicated to Professor Benjamin Weiss on the occasion of his 60th birthday.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Levin, M.B., Merzbach, E. Central limit theorems for the ergodic adding machine. Isr. J. Math. 134, 61–92 (2003). https://doi.org/10.1007/BF02787403

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02787403

Keywords

Search

Navigation