We consider statistical inference for processes with time-varying means and non-stationary time dependent errors by using wavelet analysis. First we establish an asymptotic distributional theory for the wavelet coefficients based on the strong invariance principle for non-stationary time series. When the mean function is Lipschitz continuous, its simultaneous confidence band is constructed from the asymptotic distribution of the wavelet thresholding estimators. For processes with structural breaks, inference for the breaks and empirical confidence intervals for the break locations are discussed. For broader function spaces such as the Besov space, we construct confidence bands from the thresholding estimators by using confidence intervals of the L2 risks.
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