# Spectral density

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of a stationary stochastic process or of a homogeneous random field in -dimensional space

The Fourier transform of the covariance function of a stochastic process which is stationary in the wide sense (cf. Stationary stochastic process; Random field, homogeneous). Stationary stochastic processes and homogeneous random fields for which the Fourier transform of the covariance function exists are called processes with a spectral density.

Let be an -dimensional stationary stochastic process, and let be its spectral representation ( is the spectral measure corresponding to the -th component of the multi-dimensional stochastic process ). The range of integration is in the case of discrete time , and in the case of continuous time . The process has a spectral density if all the elements of the spectral measure are absolutely continuous and if In particular, if the relation holds for , where is the covariance function of , then has a spectral density and  The situation is similar in the case of processes in continuous time . The spectral density is sometimes called the second-order spectral density, in contrast to higher spectral densities (see Spectral semi-invariant).

A homogeneous -dimensional random field has a spectral density if its spectral resolution possesses the property that its mixed derivative exists almost-everywhere, and then and How to Cite This Entry:
Spectral density. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spectral_density&oldid=13166
This article was adapted from an original article by I.G. Zhurbenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article