مرکز منطقه ای اطلاع رساني علوم و فناوري

Abstract :

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.