  We propose a difference-based nonparametric methodology for the estimation
and inference of the time-varying auto-covariance functions of a locally
stationary time series when it is contaminated by a complex trend with both
abrupt and smooth changes. Simultaneous confidence bands (SCB) with
asymptotically correct coverage probabilities are constructed for the
auto-covariance functions under complex trend. A simulation-assisted
bootstrapping method is proposed for the practical construction of the SCB.
Detailed simulation and a real data example round out our presentation.
