Confidence surfaces for the mean of locally stationary functional time series

The problem of constructing a simultaneous confidence band for the mean function of a locally stationary functional time series $ \{ X_{i,n} (t) \}_{i = 1, \ldots, n}$ is challenging as these bands can not be built on classical limit theory. On the one hand, for a fixed argument $t$ of the functions $ X_{i,n}$, the maximum absolute deviation between an estimate and the time dependent regression function exhibits (after appropriate standardization) an extreme value behaviour with a Gumbel distribution in the limit. On the other hand, for stationary functional data, simultaneous confidence bands can be built on classical central theorems for Banach space valued random variables and the limit distribution of the maximum absolute deviation is given by the sup-norm of a Gaussian process. As both limit theorems have different rates of convergence, they are not compatible, and a weak convergence result, which could be used for the construction of a confidence surface in the locally stationary case, does not exist. In this paper we propose new bootstrap methodology to construct a simultaneous confidence band for the mean function of a locally stationary functional time series, which is motivated by a Gaussian approximation for the maximum absolute deviation. We prove the validity of our approach by asymptotic theory, demonstrate good finite sample properties by means of a simulation study and illustrate its applicability analyzing a data example.