An Erdös-Rényi law for mixing processes

• Autorzy: Denker M. , Kabluchko Z.
• Języki publikacji: EN

Abstrakty

EN

We prove a large deviation type result for ψ-mixing processes and derive an ergodie version of the Erdøs-Rényi law. The result applies to expanding and Gibbs-Markov dynamical systems, including Gibbs measures and continued fractions.

Słowa kluczowe

EN

Erdös-Rényi law; large deviation; Gibbs-Markov map

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2007
• Tom: Vol. 27, Fasc. 1
• Strony: 139--149
• Opis fizyczny: Bibliogr. 17 poz.

Twórcy

autor

Denker M.

• Georg-August-Universität Göttingen, Institut für Mathematische Stochastik, Maschmühlenweg 8-10, Göttingen, Germany

autor

Kabluchko Z.

• Georg-August-Universität Göttingen, Institut für Mathematische Stochastik, Maschmühlenweg 8-10, Göttingen, Germany

Bibliografia

• [1] J. Aaronsoti and M. Denker, The Poincare series of C\Z, Ergodic Theory Dynam. Systems 19 (1999), pp. 1-20,
• [2] J. Aaronson and M. Denker, Local limit theorems for partial sums of stationary sequences generated by Gibbs-Markov maps, Stoch. Dyn. 1 (2001), pp. 193-237.
• [3] J. Aaronson, M. Denker and M. Urbański, Ergodic theory for Markov fibred systems and parabolic rational maps, Trans. Amer. Math. Soc. 337 (1993), pp. 495-548.
• [4] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math, No 470, Springer, 1975.
• [5] R. Bradley, On the ψ-mixing condition for stationary random sequences, Trans. Amer. Math. Soc. 276 (1983), pp. 55-66.
• [6] J.-R. Chazottes and P. Collet, Almost sure central limit theorems and the Erdös-Rényi law for expanding maps of the interval, Ergodic Theory Dynam. Systems 25 (2005), pp. 419-441.
• [7] M. Denker, Large deviations and the pressure function, in: Transactions of the 11th Prague Conference on Information Theory, Statistical Decision Functions, Random Processes; Prague 1990, Academia Publ. House of the Czechoslovak Acad, Science, 1992, pp. 21-33.
• [8] M. Denker, C. Grillenberger and К, Sigmund, Ergodic Theory on Compact Spaces, Lecture Notes in Math. No 527, Springer, 1976.
• [9] R. S. Ellis, Entropy, Large Deviations and Statistical Mechanics, Grundlehren 271, Springer, 1985.
• [10] P. Erdös and A. Rényi, On a new law of large numbers, J. Anal. Math. 23 (1970), pp. 103-111.
• [11] J. Grigull, Große Abweichungen und Fluktuationen für Gleichgewichtsmaße rationaler Abbil- dungen, Dissertation, Göttingen University, 1993.
• [12] S. Orey, Large deviation in ergodic theory, in: Seminar on Stochastics. Proceedings 1984, Birkhäuser, 1986, pp. 195-248.
• [13] Y. Такahashi, Entropy functional (free energy) for dynamical systems and their random perturbations, in: Proceedings of the Taniguchi Symposium on Stochastical Analysis at Katata and Kyoto, 1982, Kinokuniga Tohyo, North Holland, Amsterdam 1982.
• [14] Y. Takahashi, Two aspects of large deviation theory for large time, in: Probabilistic Methods in Mathematical Physics (Katata-Kyoto, 1985), Academic Press, Boston, MA, 1987, pp. 363-384.
• [15] W. Bryc, Large deviation by the asymptotic value method, in: Proceedings of the Conference on . Diffusion Processes, M. Punsky (Ed.), Birkhäuser, Boston, MA, 1990, pp. 447-472.
• [16] W. Bryc, On large deviations for uniformly strong mixing sequences, Stochastic Process. Appl. 41 (1992), pp. 191-202.
• [17] R. Kiesel and U. Stadtmüller, Erdös-Rényi-Shepp laws for dependent random variables, Studia Sci. Math. Hungar. 34 (1998), pp. 253-259.