Gaussian Approximation for Lag-Window Estimators and the Construction of Confidence bands for the Spectral Density

In this paper we consider the construction of simultaneous confidence bands for the spectral density of a stationary time series using a Gaussian approximation for classical lag-window spectral density estimators evaluated at the set of all positive Fourier frequencies. The Gaussian approximation opens up the possibility to verify asymptotic validity of a multiplier bootstrap procedure and, even further, to derive the corresponding rate of convergence. A small simulation study sheds light on the finite sample properties of this bootstrap proposal.