Sieve bootstrap for functional time series

A bootstrap procedure for functional time series is proposed which exploits a general vector autoregressive representation of the time series of Fourier coefficients appearing in the Karhunen-Lo\`eve expansion of the functional process. A double sieve-type bootstrap method is developed which avoids the estimation of process operators and generates functional pseudo-time series that appropriately mimic the dependence structure of the functional time series at hand. The method uses a finite set of functional principal components to capture the essential driving parts of the infinite dimensional process and a finite order vector autoregressive process to imitate the temporal dependence structure of the corresponding vector time series of Fourier coefficients. By allowing the number of functional principal components as well as the autoregressive order used to increase to infinity (at some appropriate rate) as the sample size increases, a basic bootstrap central limit theorem is established which shows validity of the bootstrap procedure proposed for functional finite Fourier transforms. Some numerical examples illustrate the good finite sample performance of the new bootstrap method proposed.