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Testing for harmonizability

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Let R(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} where G(x,y) is continuous to the right in both variables and is of bounded variation in the plane; then R(s,t) is harmonizable in that G(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, if X(t) is a real Gaussian harmonizable process, then X^n (t) is harmonizable. If X(t) is harmonizable, d^2 G(x,y) has bounded support and g(t) is a Fourier-Stieltjes transform, then X(g(t)) and X(t + g(t)) are harmonizable. If begin{equation} X(t) =int_{-infty}^{infty} f(t,u) dZ(u) end{equation} where f (t,u) = f (t - u) is a Fourier-Stieltjes transform and G(u,v) = E(Z(u)Z^{\ast } (v)) has finite total variation, then X(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.

Published in:

IEEE Transactions on Information Theory  (Volume:19 ,  Issue: 3 )