We present a general theory to quantify the uncertainty from imposing structural assumptions on the second-order structure of nonstationary Hilbert space-valued processes, which can be measured via functionals of time-dependent spectral density operators. The second-order dynamics are well-known to be elements of the space of trace-class operators, the latter is a Banach space of type 1 and of cotype 2, which makes the development of statistical inference tools more challenging. A part of our contribution is to obtain a weak invariance principle as well as concentration inequalities for (functionals of) the sequential time-varying spectral density operator. In addition, we introduce deviation measures in the nonstationary context, and derive estimators that are asymptotically pivotal. We then apply this framework and propose statistical methodology to investigate the validity of structural assumptions for nonstationary response surface data, such as low-rank assumptions in the context of time-varying dynamic fPCA and principle separable component analysis, deviations from stationarity with respect to the square root distance, and deviations from zero functional canonical coherency.

翻译：我们提出了一种通用理论，用于量化对非平稳希尔伯特空间值过程的二阶结构施加结构假设时产生的不确定性，这种不确定性可通过时间依赖谱密度算子的泛函进行测量。二阶动力学已知是迹类算子空间的元素，而该空间属于类型1与余类型2的巴拿赫空间，这使得统计推断工具的研发更具挑战性。我们贡献的一部分在于获得了序贯时变谱密度算子（的泛函）的弱不变原理及浓度不等式。此外，我们在非平稳背景下引入了偏差度量，并推导出渐近枢轴估计量。我们随后应用该框架提出统计方法，以检验非平稳响应曲面数据的结构假设有效性——包括时变动态fPCA和主可分离成分分析中的低秩假设、基于平方根距离的平稳性偏差，以及零泛函典型相干性的偏差。