On the asymptotic distribution of the periodograms for the discrete time harmonizable simple processes

Simple harmonizable processes, introduced by Soltani and Parvardeh (Theory Probab Appl 50(3):448–462, 2006), form a fairly large class of second order processes that includes stationary processes and periodically correlated processes. The spectral density of a simple process is supported by certain curves in [0,2π)2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0,2\pi )^2$$\end{document}. In this article we proceed to the inference for the spectral density of simple processes, including estimation of the spectral density supporting curves and derivation of the asymptotic distribution of the periodogram. We also introduce the “spectral cipher” that highlights active frequencies of a given time series. Theoretical derivations are exhibited using real and simulated data.