Rates of convergence in CLT and ASIP for sequences of expanding maps

TL;DR

This paper establishes rates of convergence in the Central Limit Theorem and Almost Sure Invariance Principle for sums involving sequences of expanding maps, extending classical results to non-stationary dynamical systems.

Contribution

It provides new Berry-Esseen and ASIP rates for non-stationary sequences of expanding maps and functions with bounded variation, generalizing known results for single maps.

Findings

Proves Berry-Esseen theorems with rates for non-stationary sequences.

Establishes almost sure invariance principles with convergence rates.

Extends results to maps near Axiom A maps and Hölder functions.

Abstract

We prove Berry-Esseen theorems and the almost sure invariance principle with rates for partial sums of the form $S_n=\sum_{j=0}^{n-1}f_j\circ T_{j-1}\circ\cdots\circ T_1\circ T_0$ where $f_j$ are functions with uniformly bounded ``variation" and $T_j$ is a sequence of expanding maps. Using symbolic representations similar result follow for maps $T_j$ in a small $C^1$ neighborhood of an Axiom A map and H\"older continuous functions $f_j$. All of our results are already new for a single map $T_j=T$ and a sequence of different functions $(f_j)$.