Estimation for Almost Periodic Processes

Processes with almost periodic covariance functions have spectral mass on lines parallel to the diagonal in the two-dimensional spectral plane. Methods have been given for estimation of spectral mass on the lines of spectral concentration if the locations of the lines are known. Here methods for estimating the intercepts of the lines of spectral concentration in the Gaussian case are given under appropriate conditions. The methods determine rates of convergence sufficiently fast as the sample size n → ∞ so that the spectral estimation on the estimated lines can then proceed effectively. This task involves bounding the maximum of an interesting class of non-Gaussian possibly non-stationary processes. 1. Introduction. The main objective of this paper is to present a constructive method to determine the lines of support of spectra or, equivalently, the frequencies or periods of Gaussian harmonizable processes with almost periodic covariances under appropriate conditions. The processes considered provide an interesting set of random processes that are generally not transient and not stationary and for which Fourier methods of analysis are helpful and meaningful. The lines of support of spectra are lines parallel to the diagonal in the two-dimensional spectral plane. A number of papers have discussed spectral estimation in this context but they all assume knowledge of the lines of support of the spectra. The novelty of this paper is in the presentation of constructive methods for estimating the lines of support under appropriate conditions with a rate of convergence good enough to imply bias and covariance of spectral estimation using the estimated support lines with the same asymptotic properties as if the actual lines of support were known precisely.