Explicit conditions for the CLT and related results for non-uniformly partially expanding random dynamical systems via effective RPF rates

Abstract

The purpose of this paper is to provide a first class of explicit sufficient conditions for the central limit theorem and related results in the setup of non-uniformly (partially) expanding non iid random transformations, considered as stochastic processes together with some random Gibbs measure. More precisely, we prove a central limit theorem (CLT), an almost sure invariance principle, a moderate deviations principle, Berry-Esseen type estimates and a moderate local central limit theorem for random Birkhoff sums generated by a non-uniformly partially expanding dynamical systems Tω and a random Gibbs measure μω, corresponding to a random potential ϕω with a sufficiently regular variation. In the partially expanding case the maps we consider are similar to the ones in [44], with the exception that the amount of expansion dominates the amount of contraction fiberwise and not only on the average, and with an additional regularity condition on a certain type of local variation of ϕω along inverse branches of Tω. A notable example when the maps are truly partially-expanding is the case when ϕω≡0 which corresponds to random measures of maximal entropy μω, but any potential with a sufficiently small (fiberwise) variation can be considered. Our results in the partially expanding case are new even in the uniformly random case, where all the random variables describing the maps are uniformly controlled. For properly expanding maps (as in [42,33]), the above local regularity condition allows applications also in the smooth case where the Gibbs measure is absolutely continuous with respect to the underlying volume measure and ϕω=−ln⁡JTω. For instance, we can consider certain fiberwise piecewise C2-perturbations of piecewise linear or affine maps. All of the above is achieved by first proving random, real and complex, Ruelle-Perron-Frobenius (RPF) theorems with rates that can be expressed analytically by means of certain random parameters that describe the maps (such rates will be referred to as “effective”). Using these effective rates, our conditions for the limit theorems involve some weak type of upper mixing conditions on the driving system (base map) and some integrability conditions on the norm of the random function generating the Birkhoff sums. A big part of the proof of the moderate deviations, the Berry-Esseen type estimates and the local CLT is to show how Rugh's theory [43] of complex cones contractions applies to the cones considered in [16] (and their random versions in [44]), which is new even for deterministic dynamical systems T, and in that case it yields explicit estimates on the spectral gap of appropriate deterministic complex perturbations of the transfer operator of T, as well as explicit constants in the corresponding Berry-Esseen theorem for deterministic partially expanding dynamical systems.