A vector-valued almost sure invariance principle for random expanding on average cocycles

Abstract We obtain a quenched vector-valued almost sure invariance principle (ASIP) for random expanding on average cocycles. This is achieved by combining the adapted version of Gouëzel’s approach for establishing ASIP (developed in [14]) and the recent construction of the so-called adapted norms (carried out in [15]), which in some sense eliminate the non-uniformity of the decay of correlations. For realvalued observables, we also show that the martingale approximation technique is applicable in our setup, and that it yields the ASIP with better error rates. Finally, we present an example showing the necessity of the scaling condition (10), answering a question of [15].

[1]  Matthew Nicol,et al.  A vector-valued almost sure invariance principle for hyperbolic dynamical systems , 2006, math/0606535.

[2]  V. Araújo.  Random Dynamical Systems , 2006, math/0608162.

[3]  W. Philipp,et al.  Almost sure invariance principles for partial sums of weakly dependent random variables , 1975 .

[4]  G. Froyland,et al.  A Spectral Approach for Quenched Limit Theorems for Random Expanding Dynamical Systems , 2017, 1705.02130.

[5]  A. Török,et al.  Almost sure invariance principle for sequential and non-stationary dynamical systems , 2014, 1406.4266.

[6]  G. Froyland,et al.  Almost sure invariance principle for random piecewise expanding maps , 2016, 1611.04003.

[7]  S. Gouëzel,et al.  ALMOST SURE INVARIANCE PRINCIPLE FOR DYNAMICAL SYSTEMS BY SPECTRAL METHODS , 2009, 0907.1404.

[8]  Mark F. Demers,et al.  A Functional Analytic Approach to Perturbations of the Lorentz Gas , 2012, 1210.1261.

[9]  Philip T. Maker.  The ergodic theorem for a sequence of functions , 1940 .

[10]  Davor Dragicevic,et al.  Almost Sure Invariance Principle for Random Distance Expanding Maps with a Nonuniform Decay of Correlations , 2021, Lecture Notes in Mathematics.

[11]  C. Gonz'alez-Tokman,et al.  A semi-invertible operator Oseledets theorem , 2011, Ergodic Theory and Dynamical Systems.

[12]  Gary Froyland,et al.  Thermodynamic Formalism for Random Weighted Covering Systems , 2020, Communications in Mathematical Physics.

[13]  F. Merlevède,et al.  Strong invariance principles with rate for "reverse" martingales and applications , 2012, 1209.3677.

[14]  Mikko Stenlund,et al.  A coupling approach to random circle maps expanding on the average , 2013, 1306.2942.

[15]  Mariusz Urba'nski,et al.  Distance Expanding Random Mappings, Thermodynamical Formalism, Gibbs Measures and Fractal Geometry , 2008, 0805.4580.

[16]  Ian Melbourne,et al.  Decay of correlations, central limit theorems and approximation by Brownian motion for compact Lie group extensions , 2003, Ergodic Theory and Dynamical Systems.

[17]  Davor Dragivcevi'c,et al.  Quenched limit theorems for expanding on average cocycles , 2021 .

[18]  Matthew Nicol,et al.  Communications in Mathematical Physics Almost Sure Invariance Principle for Nonuniformly Hyperbolic Systems , 2022 .

[19]  YEOR HAFOUTA.  Limit theorems for some skew products with mixing base maps , 2021, Ergodic Theory and Dynamical Systems.

[20]  C. Liverani.  Decay of correlations , 1995 .

[21]  I. Melbourne,et al.  Martingale–coboundary decomposition for families of dynamical systems , 2016, Annales de l'Institut Henri Poincaré C, Analyse non linéaire.

[22]  S. Riedel,et al.  Oseledets Splitting and Invariant Manifolds on Fields of Banach Spaces , 2019, Journal of Dynamics and Differential Equations.

[23]  J. Dedecker,et al.  Rates in almost sure invariance principle for quickly mixing dynamical systems , 2018, Stochastics and Dynamics.

[24]  S. Vaienti,et al.  Annealed and quenched limit theorems for random expanding dynamical systems , 2013, 1310.4359.

[25]  J. Buzzi.  Exponential Decay of Correlations for Random Lasota–Yorke Maps , 1999 .

[26]  THERMODYNAMIC FORMALISM FOR RANDOM TRANSFORMATIONS REVISITED , 2008 .

[27]  Y. Kifer.  Limit theorems for random transformations and processes in random environments , 1998 .

[28]  Y. Hafouta,et al.  Almost sure invariance principle for random dynamical systems via Gouëzel's approach , 2019, Nonlinearity.

[29]  M. Pollicott,et al.  Invariance Principles for Interval Maps with an Indifferent Fixed Point , 2002 .

[30]  A Vector-Valued Almost Sure Invariance Principle for Sinai Billiards with Random Scatterers , 2012, 1210.0902.

[31]  A. Korepanov.  Equidistribution for Nonuniformly Expanding Dynamical Systems, and Application to the Almost Sure Invariance Principle , 2017, 1701.03652.