This study develops an asymptotic theory for estimating the time-varying characteristics of locally stationary functional time series. We investigate a kernel-based method to estimate the time-varying covariance operator and the time-varying mean function of a locally stationary functional time series. 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 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：这项研究为估计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。我们调查了一种内核法,以估计当地固定功能时间序列的时间变化共变操作员的时间变化和时间变化平均函数。特别是,我们从共变操作员及其相关源值和元功能的内核测量员的汇合率中得出一个总合率,并为以内核为基础的当地加权样本的中值定出一个中心限值。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功能平等性的方法。