Correlation functions of continuous-time periodically correlated processes can be represented by a Fourier series with coefficient functions. It is shown that the usual estimator for stationary covariances, formed from a single sample path of the process, can be simply modified to provide a consistent (in quadratic mean) estimator for any of the coefficient functions resulting from the aforementioned representation. It is shown that, if the process is Gaussian and B/sub k/( tau ) is a Fourier integral with respect to a density function g/sub k/( lambda ), a two-dimensional periodogram, formed from a single sample function, can be smoothed along a line of constant difference frequency to provide a consistent estimator for g/sub k/( lambda ). This natural extension of the well-known procedure for stationary processes provides a method for nonparametric spectral analysis of periodically correlated processes.>

LetR(s,t)be a covariance function having the representation \begin{equation} R(s,t) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \exp (isx - ity)d^2 G(x,y) \end{equation} whereG(x,y)is continuous to the right in both variables and is of bounded variation in the plane; thenR(s,t)is harmonizable in thatG(x,y)is also a covariance. We give examples in which this result is used to determine the harmonizability of new processes and covariances that are formed by operations on old processes and covariances. Specifically, ifX(t)is a real Gaussian harmonizable process, thenX^n (t)is harmonizable. IfX(t)is harmonizable,d^2 G(x,y)has bounded support andg(t)is a Fourier-Stieltjes transform, thenX(g(t))andX(t + g(t))are harmonizable. If \begin{equation} X(t) =\int_{-\infty}^{infty} f(t,u) dZ(u) \end{equation} wheref (t,u) = f (t - u)is a Fourier-Stieltjes transform andG(u,v) = E(Z(u)Z^{\ast} (v))has finite total variation, thenX(t)is harmonizable. We also obtain a sufficient condition for Priestley's oscillatory processes to be harmonizable. We find that the Bochner-Eberlein characterization of Fourier-Stieltjes transforms, while not the only method, is particularly convenient for determining the harmonizability of these examples.