On the asymptotic normality of kernel estimators of the long run covariance of functional time series

TL;DR

This paper proves that kernel estimators of the long run covariance for stationary functional time series are asymptotically normal in $L^2$, under weak dependence conditions, enabling joint asymptotic analysis of functional principal components.

Contribution

It establishes the asymptotic normality of kernel estimators for the long run covariance in functional time series under broad dependence assumptions, including Bernoulli shift models.

Findings

Kernel estimators are asymptotically normal in $L^2$.

Joint asymptotics show asymptotic independence of functional principal components.

Results apply to most stationary functional time series models.

Abstract

We consider the asymptotic normality in $L^2$ of kernel estimators of the long run covariance kernel of stationary functional time series. Our results are established assuming a weakly dependent Bernoulli shift structure for the underlying observations, which contains most stationary functional time series models, under mild conditions. As a corollary, we obtain joint asymptotics for functional principal components computed from empirical long run covariance operators, showing that they have the favorable property of being asymptotically independent.