Normal covariances. (English) Zbl 0641.60043

Covariances R(s,t) are called normal if they can be written in the form \[ R(s,t)=\int^{\infty}_{-\infty}\int^{\infty}_{- \infty}e^{\lambda (s+t)}e^{i\mu (s-t)}dF(\lambda,\mu),\quad (s,t)\in R^ 2. \] Some properties and characteristics of normal covariances are proved (in addition to previous results of the author): 1) they are continuous on \(R^ 2\); 2) they can be characterized as a function which is nonnegative definite in some sense; 3) they can be characterized using the corresponding reproducing kernel Hilbert space.
Reviewer: T.Cipra


60G10 Stationary stochastic processes
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[1] J. Michálek: Locally stationary covariances. Trans. Tenth Prague Conf. on Inform. Theory, Statist. Dec. Funct. Random Processes, Academia, Prague 1987.
[2] J. Michálek: Random sequences with normal covariances. Kybernetika 23 (1986), 6, 443-457. · Zbl 0632.60031
[3] R. A. Silverman: Locally stationary random processes. IRE Trans. Inform. Theory IT-3 (1957), 3, 182-187.
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