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

Estimation and Inference of Time-Varying Auto-Covariance under Complex Trend: A Difference-based Approach