On the asymptotic distribution for the periodograms of almost periodically correlated (cyclostationary) processes

Almost periodically correlated (APC) processes have almost periodic mean and auto-covariance functions. These processes have spectral mass on lines parallel to the diagonal, Tj(x)=x±αj, j=1,2,…, in the two-dimensional spectral plane [0,2π)2, and contain stationary and periodically correlated processes. The aim of this article is to make almost periodically correlated processes practical in time series modeling. The main results are on the asymptotic unbiasedness of the periodogram, and the limiting distribution for the finite Fourier transform and the periodogram of APC processes. First, we put light on APC processes in more details. The periodogram is introduced and by using an auxiliary operator, we prove that the limiting distribution of the finite Fourier transform and the periodogram are multivariate complex normal and complex Wishart distributions, respectively. Finally, the accuracy of theoretical results is investigated through simulation study.