A central limit theorem for time-dependent dynamical systems

TL;DR

This paper establishes a central limit theorem for time-dependent dynamical systems, specifically for non-random, uniformly expanding maps, by identifying conditions for convergence to the normal distribution of scaled partial sums.

Contribution

It introduces new conditions ensuring CLT for non-random, time-dependent systems, including specific examples not derived from small perturbations.

Findings

Convergence to normal distribution under specified conditions

Identification of conditions for variance divergence

First examples of non-perturbed systems satisfying CLT

Abstract

The work [8] established memory loss in the time-dependent (non-random) case of uniformly expanding maps of the interval. Here we find conditions under which we have convergence to the normal distribution of the appropriately scaled Birkhoff-like partial sums of appropriate test functions. A substantial part of the problem is to ensure that the variances of the partial sums tend to infinity (cf. the zero-cohomology condition in the autonomous case). In fact, the present paper is the first one where non-random, i. e. specific examples are also found, which are not small perturbations of a given map. Our approach uses martingale approximation technique in the form of [9].