Open Access
2022 Finite sample theory for high-dimensional functional/scalar time series with applications
Qin Fang, Shaojun Guo, Xinghao Qiao
Author Affiliations +
Electron. J. Statist. 16(1): 527-591 (2022). DOI: 10.1214/21-EJS1960


Statistical analysis of high-dimensional functional times series arises in various applications. Under this scenario, in addition to the intrinsic infinite-dimensionality of functional data, the number of functional variables can grow with the number of serially dependent observations. In this paper, we focus on the theoretical analysis of relevant estimated cross-(auto)covariance terms between two multivariate functional time series or a mixture of multivariate functional and scalar time series beyond the Gaussianity assumption. We introduce a new perspective on dependence by proposing functional cross-spectral stability measure to characterize the effect of dependence on these estimated cross terms, which are essential in the estimates for additive functional linear regressions. With the proposed functional cross-spectral stability measure, we develop useful concentration inequalities for estimated cross-(auto)covariance matrix functions to accommodate more general sub-Gaussian functional linear processes and, furthermore, establish finite sample theory for relevant estimated terms under a commonly adopted functional principal component analysis framework. Using our derived non-asymptotic results, we investigate the convergence properties of the regularized estimates for two additive functional linear regression applications under sparsity assumptions including functional linear lagged regression and partially functional linear regression in the context of high-dimensional functional/scalar time series.

Funding Statement

Shaojun Guo was partially supported by the National Natural Science Foundation of China (No. 11771447).


We are grateful to the editor, the associate editor and two referees for their insightful comments, which have led to significant improvement of our paper.


Download Citation

Qin Fang. Shaojun Guo. Xinghao Qiao. "Finite sample theory for high-dimensional functional/scalar time series with applications." Electron. J. Statist. 16 (1) 527 - 591, 2022.


Received: 1 August 2020; Published: 2022
First available in Project Euclid: 10 January 2022

Digital Object Identifier: 10.1214/21-EJS1960

Primary: 62M10 , 62R10
Secondary: 62J07

Keywords: Cross-spectral stability measure , Functional linear regression , functional principal component analysis , non-asymptotics , Sparsity , sub-Gaussian functional linear process

Vol.16 • No. 1 • 2022
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