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

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

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: nicol@math.uh.edu)
FELIPE PEREZ PEREIRA
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, UK (e-mail: fp16987@bristol.ac.uk)
ANDREW TÖRÖK
Affiliation:
Department of Mathematics, University of Houston, Houston, TX77204, USA Institute of Mathematics of the Romanian Academy, Bucharest, Romania (e-mail: torok@math.uh.edu)
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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
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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