\indent Covariance matrix is of fundamental importance in many aspects of statistics. Recently, there is a surge of interest on regularized covariance matrix estimation using banding, tapering and thresholding methods, in high dimensional statistical inference, where multiple iid copies of the random vector from the underlying multivariate distribution are required. \indent In the context of time series analysis, however, it is typical that only one realization is available, so the current results are not applicable. In this talk, we shall exploit the connection between covariance matrices and spectral density functions using the idea in Toeplitz~(1911) and Grenander and Szeg\"o~(1958)