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

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.

💡 Research Summary

This paper addresses a fundamental gap in the theory of high‑dimensional functional time series: the lack of non‑asymptotic guarantees for covariance and spectral density estimation when the data are non‑Gaussian, nonlinear, and possibly far higher in dimension than the sample size. The authors introduce a pair of functional dependence measures—Φₓ(q,α) and Mₓ(q,α)—that quantify temporal dependence by coupling each observation with an independent copy of its innovation and measuring the L₂ distance between the resulting functional curves. These measures extend the scalar “physical dependence” framework of Zhang and Wu (2021) to infinite‑dimensional Hilbert spaces, while remaining tractable: explicit upper bounds can be derived for a broad class of stationary functional processes, including linear, moving‑average, and nonlinear models.

Using these dependence measures, the authors develop Nagaev‑type concentration inequalities for Hilbert‑space‑valued random elements under weak moment conditions (q > 4) and a decay parameter α > 0. The resulting bounds are uniform over all p components and hold for the element‑wise maximum norm. Consequently, the sample autocovariance operator Σ̂ₕ and the spectral density operator f̂(θ) satisfy
‖Σ̂ₕ − Σₕ‖∞ = Oₚ(√(log p / n)) and
‖f̂(θ) − f(θ)‖∞ = Oₚ(√(log p / n)),
even when p≫n. The proofs rely on martingale inequalities in general Banach spaces and careful handling of the infinite‑dimensional structure.

Two concrete applications illustrate the practical impact of these results. First, in dynamic functional principal component analysis (dynamic FPCA), the spectral density operator replaces the static covariance operator, preserving temporal dependence in the dimension‑reduction step. The authors show that the estimated eigenvalues and eigenfunctions of the spectral density converge at the same √(log p / n) rate, providing explicit error bounds for subsequent regularized procedures. Second, they consider sparse estimation of the spectral density matrix under a functional sparsity assumption: many pairs of functional series are uncorrelated across all lags. By applying a thresholding operator to the estimated spectral density, they obtain a consistent estimator whose element‑wise error is again bounded by a term proportional to √(log p / n). This enables reliable identification of zero cross‑spectral entries.

Recognizing that functional data are often observed discretely and contaminated with measurement error, the paper extends the theory to a realistic setting. Using local linear smoothing, the authors construct estimators of the autocovariance and spectral density from noisy, discretized observations. They prove that the smoothing bias (of order h², with bandwidth h) adds only a negligible term to the concentration bound, preserving the √(log p / n) convergence rate.

Extensive Monte‑Carlo simulations across Gaussian, heavy‑tailed, and nonlinear data generating mechanisms confirm the theoretical rates. The simulations demonstrate that the proposed estimators achieve the predicted error decay, that dynamic FPCA benefits from the spectral‑density‑based reduction, and that the sparse spectral estimator accurately recovers the underlying sparsity pattern.

In summary, the paper makes four major contributions: (1) novel, tractable functional dependence measures for high‑dimensional functional time series; (2) systematic Nagaev‑type non‑asymptotic concentration bounds for covariance and spectral density estimators under weak moment and dependence conditions; (3) concrete methodological advances in dynamic FPCA and sparse spectral density estimation; and (4) a rigorous treatment of discretely observed, noisy functional data. These results substantially broaden the toolbox for high‑dimensional functional time‑series analysis, opening avenues for further work on functional regression, graphical modeling, and online inference in complex, high‑frequency functional environments.