Abstract: In this article, we introduce a weighted periodogram in the class of smoothed periodograms as a consistent estimator for the spectral density matrix of a periodically correlated process. We derive its limiting distribution that appears to be a certain finite linear combination of Wishart distribution. We also provide numerical derivations for our smoothed periodogram and exhibit its asymptotic consistency using simulated data.

2010 AMS Mathematics Subject Classification: Primary: 62M10; Secondary: 62M15.

Keywords and phrases: Periodically correlated processes, spectral density matrix, Kullback–Leibler distance, smoothed periodogram, Wishart distribution, limiting distribution.