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.
Article PDF
Similar content being viewed by others
References
Aaronson, J.: An introduction to infinite ergodic theory. volume 50 of Mathematical Surveys and Monographs. American Mathematical Society, 1997
Aaronson, J., Denker, M.: A local limit theorem for stationary processes in the domain of attraction of a normal distribution. Preprint, 1998
Aaronson, J., Denker, M.: Local limit theorems for partial sums of stationary sequences generated by Gibbs-Markov maps. Stoch. Dyn. 1, 193–237 (2001)
Aaronson, J., Denker, M., Urbański, M.: Ergodic theory for Markov fibred systems and parabolic rational maps. Trans. Am. Math. Soc. 337, 495–548 (1993)
Bochner, S., Phillips, R.S.: Absolutely convergent Fourier expansions for non-commutative normed rings. Ann. Math. 43, 409–418 (1942)
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
Feller, W.: An Introduction to Probability Theory and its Applications, volume 2. Wiley Series in Probability and Mathematical Statistics. John Wiley, 1966
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)
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)
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)
Gouëzel, S.: Sharp polynomial bounds for the decay of correlations. To be published in Israel J. Math. 2002
Hennion, H.: Sur un théorème spectral et son application aux noyaux lipschitziens. Proc. Am. Math. Soc. 118, 627–634 (1993)
Holland, M.: Slowly mixing systems and intermittency maps. Preprint, 2002
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
Isola, S.: On systems with finite ergodic degree. Preprint, 2000
Ionescu-Tulcea, Marinescu, G.: Théorie ergodique pour des classes d’opérations non complètement continues. Ann. Math. 47, 140–147 (1950)
Kahane, J.-P.: Séries de Fourier absolument convergentes, volume~50 of Ergebnisse der Mathematik und ihre Grenzgebiete. Springer-Verlag, 1970
Lévy, P.: Fractions continues aléatoires. Rend. Circ. Mat. Palermo 1 (2), 170–208 (1952)
Liverani, C., Saussol, B., Vaienti, S.: A probabilistic approach to intermittency. Ergodic Theory and Dynamical Systems 19, 671–685 (1999)
Nagaev S.V.: Some limit theorems for stationary Markov chains. Theor. Probab. Appl. 2, 378–406 (1957)
Raugi, A.: Étude d’une transformation non uniformément hyperbolique de l’intervalle [0,1]. Preprint, 2002
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)
Sarig, O.: Subexponential decay of correlations. Inv. Math. 150, 629–653 (2002)
Young, L.-S.: Recurrence times and rates of mixing. Israel J. Math. 110, 153–188 (1999)
Author information
Authors and Affiliations
Corresponding author
Additional information
Mathematics Subject Classification (2000): 37A30, 37A50, 37C30, 37E05, 47A56, 60F05
Rights 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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-003-0300-4