Limit theorems for random non-uniformly expanding or hyperbolic maps with exponential tails

TL;DR

This paper establishes limit theorems such as Berry-Esseen, local CLT, and deviation principles for random non-uniformly expanding or hyperbolic maps with exponential tails, using random tower extensions and contraction properties.

Contribution

It introduces new limit theorems for a class of random dynamical systems with exponential tail properties, utilizing complex cone contraction techniques.

Findings

Proves Berry-Esseen theorem for the systems.

Establishes local central limit theorem.

Derives large and moderate deviations principles.

Abstract

We prove a Berry-Esseen theorem, a local central limit theorem and (local) large and (global) moderate deviations principles for i.i.d. (uniformly) random non-uniformly expanding or hyperbolic maps with exponential first return times. Using existing results the problem is reduced to certain random (Young) tower extensions, which is the main focus of this paper. On the random towers we will obtain our results using contraction properties of random complex equivariant cones with respect to the complex Hilbert projective metric.