This study develops an asymptotic theory for estimating the time-varying characteristics of locally stationary functional time series (LSFTS). We investigate a kernel-based method to estimate the time-varying covariance operator and the time-varying mean function of an LSFTS. In particular, we derive the convergence rate of the kernel estimator of the covariance operator and associated eigenvalue and eigenfunctions and establish a central limit theorem for the kernel-based locally weighted sample mean. As applications of our results, we discuss methods for testing the equality of time-varying mean functions in two functional samples.

翻译：本研究建立了估计局部平稳函数型时间序列（LSFTS）时变特征的渐近理论。我们探讨了基于核方法估计LSFTS时变协方差算子和时变均值函数。具体而言，我们推导了协方差算子核估计量及其关联特征值与特征函数的收敛速率，并建立了基于核的局部加权样本均值的中心极限定理。作为结果的应用，我们讨论了检验两个函数样本中时变均值函数相等性的方法。