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
Central limit theorem and stable laws for intermittent maps
Download PDF
Download PDF
  • Published: 04 November 2003

Central limit theorem and stable laws for intermittent maps

  • Sébastien Gouëzel1 

Probability Theory and Related Fields volume 128, pages 82–122 (2004)Cite this article

  • 384 Accesses

  • 102 Citations

  • Metrics details

Abstract.

In the setting of abstract Markov maps, we prove results concerning the convergence of renormalized Birkhoff sums to normal laws or stable laws. They apply to one-dimensional maps with a neutral fixed point at 0 of the form x+x 1+α, for α∈(0, 1). In particular, for α>1/2, we show that the Birkhoff sums of a Hölder observable f converge to a normal law or a stable law, depending on whether f(0)=0 or f(0)≠0. The proof uses spectral techniques introduced by Sarig, and Wiener’s Lemma in non-commutative Banach algebras.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Aaronson, J.: An introduction to infinite ergodic theory. volume 50 of Mathematical Surveys and Monographs. American Mathematical Society, 1997

  2. Aaronson, J., Denker, M.: A local limit theorem for stationary processes in the domain of attraction of a normal distribution. Preprint, 1998

  3. Aaronson, J., Denker, M.: Local limit theorems for partial sums of stationary sequences generated by Gibbs-Markov maps. Stoch. Dyn. 1, 193–237 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Aaronson, J., Denker, M., Urbański, M.: Ergodic theory for Markov fibred systems and parabolic rational maps. Trans. Am. Math. Soc. 337, 495–548 (1993)

    MathSciNet  MATH  Google Scholar 

  5. Bochner, S., Phillips, R.S.: Absolutely convergent Fourier expansions for non-commutative normed rings. Ann. Math. 43, 409–418 (1942)

    MathSciNet  MATH  Google Scholar 

  6. Dunford, N., Schwartz, J.T.: Linear Operators, Part 1: General Theory, volume~7 of Pure and Applied Mathematics: a Series of Texts and Monographs. Interscience, 1957

  7. Feller, W.: An Introduction to Probability Theory and its Applications, volume 2. Wiley Series in Probability and Mathematical Statistics. John Wiley, 1966

  8. Fisher, A., Lopes, A.O.: Exact bounds for the polynomial decay of correlation, 1/f noise and the CLT for the equilibrium state of a non-Hölder potential. Nonlinearity. 14, 1071–1104 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Guivarc’h, Y., Hardy, J.: Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d’Anosov. Ann. Inst. H. Poincaré Probab. Statist. 24, 73–98 (1988)

    MathSciNet  MATH  Google Scholar 

  10. Guivarc’h, Y., Le Jan, Y.: Asymptotic winding of the geodesic flow on modular surfaces and continued fractions. Ann. Sci. École Norm. Sup. 26 (4), 23–50 (1993)

    MATH  Google Scholar 

  11. Gouëzel, S.: Sharp polynomial bounds for the decay of correlations. To be published in Israel J. Math. 2002

  12. Hennion, H.: Sur un théorème spectral et son application aux noyaux lipschitziens. Proc. Am. Math. Soc. 118, 627–634 (1993)

    MathSciNet  MATH  Google Scholar 

  13. Holland, M.: Slowly mixing systems and intermittency maps. Preprint, 2002

  14. Hu, H.: Rates of convergence to equilibriums and decay of correlations. Announcement in the Kyoto 2002 conference available at ndds. math.sci.hokudai.ac.jp/data/NDDS/1024888882-hu.ps

  15. Isola, S.: On systems with finite ergodic degree. Preprint, 2000

  16. Ionescu-Tulcea, Marinescu, G.: Théorie ergodique pour des classes d’opérations non complètement continues. Ann. Math. 47, 140–147 (1950)

    MathSciNet  Google Scholar 

  17. Kahane, J.-P.: Séries de Fourier absolument convergentes, volume~50 of Ergebnisse der Mathematik und ihre Grenzgebiete. Springer-Verlag, 1970

  18. Lévy, P.: Fractions continues aléatoires. Rend. Circ. Mat. Palermo 1 (2), 170–208 (1952)

    Google Scholar 

  19. Liverani, C., Saussol, B., Vaienti, S.: A probabilistic approach to intermittency. Ergodic Theory and Dynamical Systems 19, 671–685 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  20. Nagaev S.V.: Some limit theorems for stationary Markov chains. Theor. Probab. Appl. 2, 378–406 (1957)

    Google Scholar 

  21. Raugi, A.: Étude d’une transformation non uniformément hyperbolique de l’intervalle [0,1]. Preprint, 2002

  22. Rousseau-Egele, J.: Un théorème de la limite locale pour une classe de transformations dilatantes et monotones par morceaux. Ann. Probab. 11, 772–788 (1983)

    MathSciNet  MATH  Google Scholar 

  23. Sarig, O.: Subexponential decay of correlations. Inv. Math. 150, 629–653 (2002)

    Article  MATH  Google Scholar 

  24. Young, L.-S.: Recurrence times and rates of mixing. Israel J. Math. 110, 153–188 (1999)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département de Mathématiques et Applications, École Normale Supérieure, 45 rue d’Ulm, 75005, Paris, France

    Sébastien Gouëzel

Authors
  1. Sébastien Gouëzel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sébastien Gouëzel.

Additional information

Mathematics Subject Classification (2000): 37A30, 37A50, 37C30, 37E05, 47A56, 60F05

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Gouëzel, S. Central limit theorem and stable laws for intermittent maps. Probab. Theory Relat. Fields 128, 82–122 (2004). https://doi.org/10.1007/s00440-003-0300-4

Download citation

  • Received: 10 December 2002

  • Revised: 06 September 2003

  • Published: 04 November 2003

  • Issue Date: January 2004

  • DOI: https://doi.org/10.1007/s00440-003-0300-4

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

  • Decay of correlations
  • Intermittency
  • Countable Markov shift
  • Central limit theorem
  • Stable laws
  • Wiener’s Lemma
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