Asymptotic Theory for Regularized Estimation in Functional Time Series Models

Abstract

Functional autoregressive (FAR) models provide a fundamental framework for analyzing temporally dependent functional data. However, the infinite-dimensional nature of the underlying Hilbert space introduces intrinsic ill-posedness, as the autocovariance operators are compact and lack bounded inverses. This paper develops a new theoretical framework for the regularized estimation and asymptotic analysis of FAR models. Leveraging Hilbert space theory, we rigorously characterize the distinction between finite- and infinite-dimensional time series analysis and formalize the necessity of regularization. To stabilize the estimation of autoregressive operators, we introduce a Tikhonov regularization scheme and derive Yule-Walker-type estimators in a general Hilbert space, and further specialize to the $L^2$ space for explicit forms. Within this unified framework, we establish the consistency and asymptotic normality of the regularized estimators and reveal that asymptotic normality can be achieved only for the predictors rather than the operator estimates themselves. Furthermore, we derive the mean squared prediction error (MSPE) and decompose its bias-variance structure. A comprehensive simulation study and an application to high-frequency functional data from wearable devices demonstrate the practical validity of the theory and the ability of FAR models to capture dynamic functional patterns.