# Sunklodas' approach to normal approximation for time-dependent dynamical systems

## TL;DR

This paper develops a normal approximation method for sums in time-dependent dynamical systems, providing explicit error bounds and applying Stein's method to systems like expanding maps and intermittent systems.

## Contribution

It introduces a new normal approximation approach for time-dependent systems with explicit error rates and constants, extending Stein's method to this context.

## Key findings

- Error in approximation decays at rates $O(N^{-1/2})$ or $O(N^{-1/2} \log N)$
- Conditions depend on the normalizing sequence $b(N)$ and metric used
- Applications include expanding maps and intermittent systems

## Abstract

We consider time-dependent dynamical systems arising as sequential compositions of self-maps of a probability space. We establish conditions under which the Birkhoff sums for multivariate observations, given a centering and a general normalizing sequence $b(N)$ of invertible square matrices, are approximated by a normal distribution with respect to a metric of regular test functions. Depending on the metric and the normalizing sequence $b(N)$, the conditions imply that the error in the approximation decays either at the rate $O(N^{-1/2})$ or the rate $O(N^{-1/2} \log N)$, under the additional assumption that $\Vert b(N)^{-1} \Vert \lesssim N^{-1/2}$. The error comes with a multiplicative constant whose exact value can be computed directly from the conditions. The proof is based on an observation due to Sunklodas regarding Stein's method of normal approximation. We give applications to one-dimensional random piecewise expanding maps and to sequential, random, and quasistatic intermittent systems.