# Bootstrapping Covariance Operators of Functional Time Series

## TL;DR

This paper proposes a bootstrap-based method for hypothesis testing on the covariance operators of functional time series, utilizing full functional data without dimension reduction, applicable to various covariance measures.

## Contribution

It introduces a bootstrap approach for covariance operator testing in dependent functional time series, extending existing methods to dependent data and multiple covariance types.

## Key findings

- Bootstrap methods effectively approximate the distribution of covariance operators.
- The approach applies to autocovariance, cross covariance, and general covariance operators.
- Simulation results demonstrate the method's accuracy and applicability.

## Abstract

For testing hypothesis on the covariance operator of functional time series, we suggest to use the full functional information and to avoid dimension reduction techniques. The limit distribution follows from the central limit theorem of the weak convergence of the partial sum process in general Hilbert space applied to the product space. In order to obtain critical values for tests, we generalize bootstrap results from the independent to the dependent case. This results can be applied to covariance operators, autocovariance operators and cross covariance operators. We discuss one sample and changepoint tests and give some simulation results.