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.

翻译：函数自回归（FAR）模型为分析时间依赖的函数数据提供了基础框架。然而，由于底层希尔伯特空间的无限维特性，自协方差算子具有紧性且缺乏有界逆算子，从而引入了固有的不适定性。本文针对FAR模型的正则化估计与渐近分析，建立了一个新的理论框架。基于希尔伯特空间理论，我们严格刻画了有限维与无限维时间序列分析之间的区别，并形式化地论证了正则化的必要性。为稳定自回归算子的估计，我们引入Tikhonov正则化方案，在一般希尔伯特空间中推导了Yule-Walker型估计量，并进一步在$L^2$空间中给出其显式形式。在此统一框架下，我们证明了正则化估计量的一致性与渐近正态性，并揭示了渐近正态性仅适用于预测变量而非算子估计本身。此外，我们推导了均方预测误差（MSPE）并分解了其偏差-方差结构。通过综合仿真研究以及对可穿戴设备高频函数数据的应用，验证了该理论的实际有效性，并展示了FAR模型捕捉动态函数模式的能力。