Fluctuation bounds for chaos plus noise in dynamical systems

TL;DR

This paper derives fluctuation bounds for statistical estimators in noisy chaotic time series, providing theoretical guarantees for measures like auto-covariance and correlation dimension.

Contribution

It introduces concentration inequality-based bounds for key statistical quantities in chaotic systems with observational noise, applicable to systems like the Hénon attractor.

Findings

Fluctuation bounds hold for auto-covariance, empirical measure, and kernel density estimators.

Results apply uniformly over all sample sizes n.

Includes systems such as the Hénon attractor with Benedicks-Carleson parameters.

Abstract

We are interested in time series of the form $y_{n} = x_{n} + \xi_{n}$ where ${x_{n}}$ is generated by a chaotic dynamical system and where $\xi_{n}$ models observational noise. Using concentration inequalities, we derive fluctuation bounds for the auto-covariance function, the empirical measure, the kernel density estimator and the correlation dimension evaluated along $y_{0}, ..., y_{n}$, for all $n$. The chaotic systems we consider include for instance the H\'{e}non attractor for Benedicks-Carleson parameters.