On estimation of mean and covariance functions in repeated time series with long-memory errors*

We consider kernel estimation of trend and covariance functions in models typically encountered in functional data analysis (FDA), with the modification that the random curves are perturbed by error processes that exhibit short- or long-range dependence. Uniform convergence of standardized maximal differences between estimated and true (trend and covariance) functions is established. For the covariance function, a transformation based on contrasts is proposed that does not require explicit trend estimation. Improved estimators can be obtained by using higher-order kernels.