# A note on quadratic forms of stationary functional time series under mild conditions

## TL;DR

This paper investigates the distributional behavior of quadratic forms in stationary functional time series, providing consistency rates for spectral density estimators and establishing joint weak convergence to Gaussian operators.

## Contribution

It introduces new methods for analyzing quadratic forms in functional time series and proves weak convergence results under mild moment conditions.

## Key findings

- Consistency rates for spectral density operator estimators
- Joint weak convergence to complex Gaussian operators
- Distributional properties of the long-run covariance operator

## Abstract

We study distributional properties of a quadratic form of a stationary functional time series under mild moment conditions. As an important application, we obtain consistency rates of estimators of spectral density operators and prove joint weak convergence to a vector of complex Gaussian random operators. Weak convergence is established based on an approximation of the form via transforms of Hilbert-valued martingale difference sequences. As a side-result, the distributional properties of the long-run covariance operator are established.