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

翻译：本文为估算当地固定功能时间序列的时间变化特性开发了一种无症状理论。 我们采用了以内核为基础的方法来估计当地固定功能时间序列的时间变化共变操作员和时间变化平均函数。 随后, 我们得出了共变操作员及相关源值和元元元的内核测量员的趋同率。 我们还为以内核为基础的当地加权抽样平均值设定了一个中心限值。 作为我们结果的应用, 我们讨论了对当地固定功能运行共变时间序列的预测, 以及在两个功能样本中测试时间变化平均函数平等性的方法 。