Marchenko–Pastur Laws for Daniell Smoothed Periodograms

ABSTRACT Given a sample from a ‐dimensional stationary time series , the most commonly used estimator for the spectral density matrix at a given frequency is the Daniell smoothed periodogram which is an average over many periodograms at slightly perturbed frequencies. We prove that the Marchenko–Pastur law holds for the eigenvalues of uniformly in , when and grow with such that and for some . This demonstrates that high‐dimensional effects can cause to become inconsistent, even when the dimension is much smaller than the sample size . Notably, we do not assume independence of the components of the time series. The Marchenko‐Pastur law thus holds for Daniell smoothed periodograms, even when it does not necessarily hold for sample auto‐covariance matrices of the same processes.