Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-09T01:33:50.020Z Has data issue: false hasContentIssue false
  • English
  • Français

Large deviations and central limit theorems for sequential and random systems of intermittent maps

Part of: Dynamical systems with hyperbolic behavior Random dynamical systems Limit theorems Ergodic theory

Published online by Cambridge University Press:  07 October 2020

MATTHEW NICOL ,
FELIPE PEREZ PEREIRA  and
ANDREW TÖRÖK
MATTHEW NICOL
Affiliation:
Department of Mathematics, University of Houston, Houston, TX77204, USA (e-mail: [email protected])
FELIPE PEREZ PEREIRA
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, UK (e-mail: [email protected])
ANDREW TÖRÖK
Affiliation:
Department of Mathematics, University of Houston, Houston, TX77204, USA Institute of Mathematics of the Romanian Academy, Bucharest, Romania (e-mail: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We obtain large and moderate deviation estimates for both sequential and random compositions of intermittent maps. We also address the question of whether or not centering is necessary for the quenched central limit theorems obtained by Nicol, Török and Vaienti [Central limit theorems for sequential and random intermittent dynamical systems. Ergod. Th. & Dynam. Sys.38(3) (2018), 1127–1153] for random dynamical systems comprising intermittent maps. Using recent work of Abdelkader and Aimino [On the quenched central limit theorem for random dynamical systems. J. Phys. A 49(24) (2016), 244002] and Hella and Stenlund [Quenched normal approximation for random sequences of transformations. J. Stat. Phys.178(1) (2020), 1–37] we extend the results of Nicol, Török and Vaienti on quenched central limit theorems for centered observables over random compositions of intermittent maps: first by enlarging the parameter range over which the quenched central limit theorem holds; and second by showing that the variance in the quenched central limit theorem is almost surely constant (and the same as the variance of the annealed central limit theorem) and that centering is needed to obtain this quenched central limit theorem.

Keywords

large deviationscentral limit theoremsstationary stochastic processesrandom dynamical systems

MSC classification

Primary: 37A50: Relations with probability theory and stochastic processes 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) 60F10: Large deviations 37H99: None of the above, but in this section
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 9 , September 2021 , pp. 2805 - 2832
Check for updates
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abdelkader, M. and Aimino, R.. On the quenched central limit theorem for random dynamical systems. J. Phys. A 49(24) (2016), 244002.CrossRefGoogle Scholar
Aimino, R., Hu, H., Nicol, M., Török, A. and Vaienti, S.. Polynomial loss of memory for maps of the interval with a neutral fixed point. Discrete Contin. Dyn. Syst. 35(3) (2015), 793–806.CrossRefGoogle Scholar
Ayyer, A., Liverani, C. and Stenlund, M.. Quenched CLT for random toral automorphism. Discrete Contin. Dyn. Syst. 24(2) (2009), 331–348.CrossRefGoogle Scholar
Aimino, R. and Freitas, J. M.. Large deviations for dynamical systems with stretched exponential decay of correlations. Port. Math. 76 (2020), 143–152.CrossRefGoogle Scholar
Aimino, R., Nicol, M. and Vaienti, S.. Annealed and quenched limit theorems for random expanding dynamical systems. Probab. Theory Related Fields 162(1–2) (2015), 233–274.CrossRefGoogle Scholar
Bahsoun, W. and Bose, C.. Corrigendum: Mixing rates and limit theorems for random intermittent maps (2016 Nonlinearity 29 1417) [MR3476513]. Nonlinearity 29(12) (2016), C4.CrossRefGoogle Scholar
Bahsoun, W. and Bose, C.. Mixing rates and limit theorems for random intermittent maps. Nonlinearity 29(4) (2016), 1417–1433.CrossRefGoogle Scholar
Bahsoun, W., Bose, C. and Ruziboev, M.. Quenched decay of correlations for slowly mixing systems. Trans. Amer. Math. Soc. 372(9) (2019), 6547–6587.CrossRefGoogle Scholar
Boyarsky, A. and P. Góra, . Invariant measures and dynamical systems in one dimension. Laws of Chaos (Probability and its Applications). Birkhäuser Boston, Inc., Boston, MA, 1997.CrossRefGoogle Scholar
Conze, J.-P. and Raugi, A.. Limit theorems for sequential expanding dynamical systems on $\left[0,1\right]$ . Ergodic Theory and Related Fields (Contemporary Mathematics, 430). American Mathematical Society, Providence, RI, 2007, pp. 89–121.CrossRefGoogle Scholar
Gordin, M. I.. The central limit theorem for stationary processes. Dokl. Akad. Nauk SSSR 188 (1969), 739–741.Google Scholar
Hella, O. and Leppänen, J.. Central limit theorems with a rate of convergence for time-dependent intermittent maps. Stoch. Dyn. 20(4) (2020), 2050025.CrossRefGoogle Scholar
Hella, O. and Stenlund, M.. Quenched normal approximation for random sequences of Transformations. J. Stat. Phys. 178(1) (2020), 1–37.CrossRefGoogle Scholar
Karr, A. F.. Probability. Springer Texts in Statistics. Springer, New York, NY, 1993.Google Scholar
Korepanov, A. and Leppänen, J.. Loss of memory and moment bounds for nonstationary intermittent dynamical systems. Preprint, 2020, arXiv:2007.07616.Google Scholar
Liverani, Carlangelo. Central limit theorem for deterministic systems. International Conference on Dynamical Systems (Montevideo, 1995) (Pitman Research Notes in Mathematics Series, 362). Longman, Harlow, UK, 1996, pp. 56–75.Google Scholar
Liverani, C., Saussol, B. and Vaienti, S.. A probabilistic approach to intermittency. Ergod. Th. & Dynam. Sys. 19(3) (1999), 671–685.CrossRefGoogle Scholar
Melbourne, I.. Large and moderate deviations for slowly mixing dynamical systems. Proc. Amer. Math. Soc. 137(5) (2009), 1735–1741.CrossRefGoogle Scholar
Melbourne, I. and Nicol, M.. Large deviations for nonuniformly hyperbolic systems. Trans. Amer. Math. Soc. 360(12) (2008), 6661–6676.CrossRefGoogle Scholar
Merlevède, F., Peligrad, M. and Utev, S.. Recent advances in invariance principles for stationary sequences. Probab. Surv. 3 (2006), 1–36.CrossRefGoogle Scholar
Nicol, M., Török, A. and Vaienti, S.. Central limit theorems for sequential and random intermittent dynamical systems. Ergod. Th. & Dynam. Sys. 38(3) (2018), 1127–1153.CrossRefGoogle Scholar
Rio, E.. Asymptotic theory of weakly dependent random processes. Probability Theory and Stochastic Modelling. Vol. 80. Springer, Berlin, 2017. Translated from the 2000 French edition.Google Scholar
Viana, M. and Oliveira, K.. Foundations of Ergodic Theory (Cambridge Studies in Advanced Mathematics, 151). Cambridge University Press, Cambridge, 2016.CrossRefGoogle Scholar