Convergence of covariance and spectral density estimates for high-dimensional functional time series

TL;DR

This paper develops non-asymptotic convergence bounds for covariance and spectral density estimates in high-dimensional, non-Gaussian functional time series, with applications to PCA and spectral density estimation.

Contribution

It introduces novel dependence measures and provides the first non-asymptotic theory for spectral density estimation in high-dimensional, non-Gaussian functional time series.

Findings

Established systematic non-asymptotic concentration bounds.

Demonstrated the effectiveness through applications to PCA and spectral density estimation.

Validated theoretical results with extensive simulations.

Abstract

Second-order characteristics including covariance and spectral density functions are fundamentally important for both statistical applications and theoretical analysis in functional time series. In the high-dimensional setting where the number of functional variables is large relative to the length of functional time series, non-asymptotic theory for covariance function estimation has been developed for Gaussian and sub-Gaussian functional linear processes. However, corresponding non-asymptotic results for high-dimensional non-Gaussian and nonlinear functional time series, as well as for spectral density function estimation, are largely unexplored. In this paper, we introduce novel functional dependence measures, based on which we establish systematic non-asymptotic concentration bounds for estimates of (auto)covariance and spectral density functions in high-dimensional and non-Gaussian settings. We then illustrate the usefulness of our convergence results through two applications to dynamic functional principal component analysis and sparse spectral density function estimation. To handle the practical scenario where curves are discretely observed with errors, we further develop convergence rates of the corresponding estimates obtained via a nonparametric smoothing method. Finally, extensive simulation studies are conducted to corroborate our theoretical findings.