  A bootstrap procedure for constructing prediction bands for a stationary
functional time series is proposed. The procedure exploits a general vector
autoregressive representation of the time-reversed series of Fourier
coefficients appearing in the Karhunen-Loeve representation of the functional
process. It generates backward-in-time, functional replicates that adequately
mimic the dependence structure of the underlying process in a model-free way
and have the same conditionally fixed curves at the end of each functional
pseudo-time series. The bootstrap prediction error distribution is then
calculated as the difference between the model-free, bootstrap-generated future
functional observations and the functional forecasts obtained from the model
used for prediction. This allows the estimated prediction error distribution to
account for the innovation and estimation errors associated with prediction and
the possible errors due to model misspecification. We establish the asymptotic
validity of the bootstrap procedure in estimating the conditional prediction
error distribution of interest, and we also show that the procedure enables the
construction of prediction bands that achieve (asymptotically) the desired
coverage. Prediction bands based on a consistent estimation of the conditional
distribution of the studentized prediction error process also are introduced.
Such bands allow for taking more appropriately into account the local
uncertainty of prediction. Through a simulation study and the analysis of two
data sets, we demonstrate the capabilities and the good finite-sample
performance of the proposed method.
