Spectral density
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
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be an -dimensional stationary stochastic process, and let
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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
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if all the elements
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of the spectral measure are absolutely continuous and if
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In particular, if the relation
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holds for ,
where
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is the covariance function of , then
has a spectral density and
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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
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and
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References
[1] | Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian) |
[2] | Yu.A. Rozanov, "Stationary random processes" , Holden-Day (1967) (Translated from Russian) |
Spectral density. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spectral_density&oldid=48758