View source for Spectral density

''of a stationary stochastic process or of a homogeneous random field in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086370/s0863701.png" />-dimensional space''

The [[Fourier transform|Fourier transform]] of the covariance function of a stochastic process which is stationary in the wide sense (cf. [[Stationary stochastic process|Stationary stochastic process]]; [[Random field, homogeneous|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086370/s0863703.png" />-dimensional stationary stochastic process, and let

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be its spectral representation (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086370/s0863705.png" /> is the [[Spectral measure|spectral measure]] corresponding to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086370/s0863706.png" />-th component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086370/s0863707.png" /> of the multi-dimensional stochastic process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086370/s0863708.png" />). The range of integration is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086370/s0863709.png" /> in the case of discrete time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086370/s08637010.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086370/s08637011.png" /> in the case of continuous time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086370/s08637012.png" />. The process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086370/s08637013.png" /> has a spectral density

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if all the elements

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of the spectral measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086370/s08637016.png" /> are absolutely continuous and if

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In particular, if the relation

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holds for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086370/s08637019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086370/s08637020.png" /> where

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is the covariance function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086370/s08637022.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086370/s08637023.png" /> has a spectral density and

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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086370/s08637025.png" /></td> </tr></table>

The situation is similar in the case of processes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086370/s08637026.png" /> in continuous time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086370/s08637027.png" />. The spectral density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086370/s08637028.png" /> is sometimes called the second-order spectral density, in contrast to higher spectral densities (see [[Spectral semi-invariant|Spectral semi-invariant]]).

A homogeneous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086370/s08637029.png" />-dimensional random field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086370/s08637030.png" /> has a spectral density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086370/s08637031.png" /> if its [[Spectral resolution|spectral resolution]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086370/s08637032.png" /> possesses the property that its mixed derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086370/s08637033.png" /> exists almost-everywhere, and then

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and

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====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.V. [Yu.V. Prokhorov] Prohorov,   Yu.A. Rozanov,   "Probability theory, basic concepts. Limit theorems, random processes" , Springer  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  Yu.A. Rozanov,   "Stationary random processes" , Holden-Day  (1967)  (Translated from Russian)</TD></TR></table>