This article explores a general factor structure for high-dimensional nonstationary functional time series, encompassing a wide range of factor models studied in the existing literature. We investigate the asymptotic spectral behaviors of the sample covariance operator under this general data structure. A novel fundamental sufficient condition, formulated in terms of a newly introduced effective rank tailored to this setup, is established under which empirical eigen-analysis yields spurious results, rendering sample eigenvalues and eigenvectors unreliable for accurately recovering the underlying factor structure. This generalizes the results of Onatski and Wang [2021] from typical high-dimensional time series (HDTS) to the more intricate functional framework. The newly defined effective rank is rigorously analyzed through a decomposition of the effects attributable to functional factor loadings and functional factors. Contrary to the findings in the HDTS setting, empirical eigen-analysis of models with only a small number of strong non-stationary factors may still produce spurious limits in the functional framework. Therefore, additional caution is warranted when applying covariance-based statistical methods to potentially nonstationary functional data. Simulation studies are performed to determine conditions under which spurious limits occur. Real data analysis on age-specific mortality rate data from multiple locations is conducted for evidence of spurious factors induced by empirical eigen-analysis.

翻译：本文探讨了一种适用于高维非平稳函数时间序列的通用因子结构，该结构涵盖了现有文献中广泛研究的多种因子模型。我们在这种通用数据结构下研究了样本协方差算子的渐近谱行为。基于针对该设定新定义的有效秩，建立了一个新颖的基础充分条件，在该条件下经验特征分析会产生虚假结果，使得样本特征值和特征向量无法可靠地用于准确恢复潜在的因子结构。这一结果将Onatski和Wang[2021]的研究从典型高维时间序列推广至更为复杂的函数框架。新定义的有效秩通过分解归因于函数因子载荷和函数因子的效应进行了严格分析。与高维时间序列设定的发现相反，仅包含少数强非平稳因子的模型的经验特征分析在函数框架下仍可能产生虚假极限。因此，在对潜在非平稳函数数据应用基于协方差的统计方法时，需要格外谨慎。通过仿真研究确定了产生虚假极限的条件。对来自多个地点的特定年龄死亡率数据进行实际数据分析，以寻找由经验特征分析引发的虚假因子存在的证据。