Electronic Journal of Probability

Decay of Correlations for Non-Hölderian Dynamics. A Coupling Approach

Xavier Bressaud, Roberto Fernandez, and Antonio Galves

Full-text: Open access

Abstract

We present an upper bound on the mixing rate of the equilibrium state of a dynamical system defined by the one-sided shift and a non Hölder potential of summable variations. The bound follows from an estimation of the relaxation speed of chains with complete connections with summable decay, which is obtained via a explicit coupling between pairs of chains with different histories.

Article information

Source
Electron. J. Probab., Volume 4 (1999), paper no. 3, 19 pp.

Dates
Accepted: 4 March 1999
First available in Project Euclid: 4 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1457125512

Digital Object Identifier
doi:10.1214/EJP.v4-40

Mathematical Reviews number (MathSciNet)
MR1675304

Zentralblatt MATH identifier
0917.58017

Subjects
Primary: 58F11
Secondary: 60G10: Stationary processes

Keywords
Dynamical systems non-Hölder dynamics mixing rate chains with complete connections relaxation speed coupling methods

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Bressaud, Xavier; Fernandez, Roberto; Galves, Antonio. Decay of Correlations for Non-Hölderian Dynamics. A Coupling Approach. Electron. J. Probab. 4 (1999), paper no. 3, 19 pp. doi:10.1214/EJP.v4-40. https://projecteuclid.org/euclid.ejp/1457125512


Export citation

References

  • M BARNSLEY, S. DEMKO, J. ELTON and J. GERINOMO (1988). Invariant measures for Markov processes arising from iterated function systems with place-dependent probabilities. Ann. Inst. Henri Poincaré, Prob. Statist., 24: 367–394. Erratum (1989) Ann. Inst. H. Poincaré Probab. Statist. 25: 589–590.
  • H. BERBEE (1987). Chains with infinite connections: Uniqueness and Markov representation. Probab. Th. Rel. Fields, 76: 243–253.
  • R. BOWEN (1975). Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, volume 470. Springer.
  • X. BRESSAUD, R. FERNANDEZ, and A. GALVES (1998). Speed of $\overline d$-convergence for Markov approximations of chains with complete connections. A coupling approach. Preprint. To be published in Stoch. Proc. and Appl.
  • Z. COELHO and A. N. QUAS (1998). Criteria for $\overline d$-continuity. Trans. Amer. Math. Soc., 350: 3257–3268.
  • K. BURDZY and W. KENDALL (1998). Efficient Markovian couplings: examples and counterexamples. Preprint.
  • W. DOEBLIN (1938). Exposé sur la théorie des chaînes simples constantes de Markoff à un nombre fini d'états. Rev. Math. Union Interbalkanique, 2: 77–105.
  • W. DOEBLIN (1940). Remarques sur la théorie métrique des fractions continues. Composition Math., 7: 353–371.
  • W. DOEBLIN and R. FORTET (1937). Sur les chaînes à liaisons complétes. Bull. Soc. Math. France, 65: 132–148.
  • T. E. HARRIS (1955). On chains of infinite order. Pacific J. Math., 5: 707–724.
  • M. IOSIFESCU (1978). Recent advances in the metric theory of continued fractions. Trans. 8th Prague Conference on Information Theory, Statistical Decision Functions, Random Processes (Prague, 1978), Vol. A, pp. 27–40, Reidel, Dordrecht-Boston, Mass.
  • M. IOSIFESCU (1992). A coupling method in the theory of dependence with complete connections according to Doeblin. Rev. Roum. Math. Pures et Appl., 37: 59–65.
  • M. IOSIFESCU and S. GRIGORESCU (1990). Dependence with Complete Connections and its Applications, Cambridge University Press, Cambridge.
  • M. IOSIFESCU and R. THEODORESCU (1969). Random Processes and Learning, Springer-Verlag, Berlin.
  • T. KAIJSER (1981). On a new contraction condition for random systems with complete connections. Rev. Roum. Math. Pures et Appl., 26: 1075–1117.
  • T. KAIJSER (1994). On a theorem of Karlin. Acta Applicandae Mathematicae, 34: 51–69.
  • S. KARLIN (1953). Some random walks arising in learning models. J. Pacific Math., 3: 725–756.
  • A. KONDAH, V. MAUME and B. SCHMITT (1996). Vitesse de convergence vers l'état d'équilibre pour des dynamiques markoviennes non Höldériennes. Technical Report 88, Université de Bourgogne.
  • S. P. LALLEY (1986). Regeneration representation for one-dimensional Gibbs states. Ann. Prob., 14: 1262–1271.
  • F. LEDRAPPIER (1974). Principe variationnel et systèmes dynamiques symboliques. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 30: 185–202.
  • T. LINDVALL (1991). W. Doeblin 1915–1940. Ann. Prob., 19: 929–934.
  • T. LINDVALL (1992). Lectures on the Coupling Method, Wiley, New York.
  • C. LIVERANI (1995). Decay of correlations. Ann. of Math., 142(2): 239–301.
  • F. NORMAN (1972). Markov Processes and Learning Models, Academic Press, New York.
  • O. ONICESCU and G. MIHOC (1935). Sur les chaînes statistiques. C. R. Acad. Sci. Paris, 200: 5121–-512.
  • O. ONICESCU and G. MIHOC (1935a). Sur les chaînes de variables statistiques. Bull. Sci., Math., 59: 174–192.
  • W. PARRY and M. POLLICOTT (1990). Zeta functions and the periodic structure of hyperbolic dynamics. Asterisque 187-188.
  • M. POLLICOTT (1997). Rates of mixing for potentials of summable variation. Preprint.
  • A. N. QUAS (1996). Non-ergodicity for $C\sp 1$ expanding maps and $g$-measures. Ergod. Th. Dynam. Sys., 16: 531–543.
  • D. RUELLE (1978). Thermodynamic formalism. Encyclopedia of Mathematics and its applications, volume 5. Addison-Wesley, Reading, Massachusetts.
  • P. WALTERS (1975). Ruelle's operator theorem and g-measures. Trans. Amer. Math. Soc., 214: 375–387.
  • P. WALTERS (1978). Invariant measures and equilibrium states for some mappings which expand distances. Trans. Amer. Math. Soc., 236: 121–153.1
  • L.-S. YOUNG (1997). Recurrence times and rates of mixing. Preprint.