Normal covariances. (English) Zbl 0641.60043

Michálek, Jiří

Kybernetika 24, No. 1, 17-27 (1988).

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

MSC:

60G10

Stationary stochastic processes

Keywords:

locally stationary covariances; properties and characteristics of normal covariances; reproducing kernel Hilbert space

References:

[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.