Asymptotic normality of sinusoidal frequencies estimated by second-order algorithms for mixed spectra time series

This paper addresses the asymptotic normal distribution of the sample covariance matrix of mixed spectra time series containing a sum of sinusoids and a linear stationary process. A new central limit theorem is proved for real or complex valued processes whose linear stationary process is possibly noncircular and not necessarily Gaussian. As an application of this result, the asymptotic normal distribution of any sinusoidal frequency estimator of such a time series based on second-order statistics is deduced. The case of the noise whitening is also considered in this general formulation. It is shown, in particular, that under mild assumptions, the asymptotic performance of most covariance-based frequency estimators is independent of the distribution of the noise.