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''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''
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''of a stationary stochastic process or of a homogeneous random field in $  n $-
 +
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.
 
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.
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Let
 
Let
  
<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/s0863702.png" /></td> </tr></table>
+
$$
 +
X ( t) = \{ X _ {k} ( t) \} _ {k=} 1  ^ {n}
 +
$$
  
be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086370/s0863703.png" />-dimensional stationary stochastic process, and let
+
be an $  n $-
 +
dimensional stationary stochastic process, and let
  
<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/s0863704.png" /></td> </tr></table>
+
$$
 +
X ( t)  = \int\limits e ^ {i t \lambda } \Phi ( d \lambda ) ,\  \Phi  = \
 +
\{ \Phi _ {k} \} _ {k=} 1  ^ {n}
 +
$$
  
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
+
be its spectral representation ( $  \Phi _ {k} $
 +
is the [[Spectral measure|spectral measure]] corresponding to the $  k $-
 +
th component $  X _ {k} ( t) $
 +
of the multi-dimensional stochastic process $  X ( t) $).  
 +
The range of integration is $  - \pi \leq  \lambda \leq  \pi $
 +
in the case of discrete time $  t $,  
 +
and $  - \infty < \lambda < + \infty $
 +
in the case of continuous time $  t $.  
 +
The process $  X ( t) $
 +
has a spectral density
  
<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/s08637014.png" /></td> </tr></table>
+
$$
 +
f ( \lambda )  = \{ f _ {k,l} ( \lambda ) \} _ {k,l=} 1  ^ {n} ,
 +
$$
  
 
if all the elements
 
if all the elements
  
<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/s08637015.png" /></td> </tr></table>
+
$$
 +
F _ {k,l} ( \Delta )  = {\mathsf E}
 +
\Phi _ {k} ( \Delta ) \overline{ {\Phi _ {l} ( \Delta ) }}\; ,\ \
 +
k , l = {1 \dots n } ,
 +
$$
  
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
+
of the spectral measure $  F = \{ F _ {k,l} \} _ {k,l=} 1  ^ {n} $
 +
are absolutely continuous and if
  
<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/s08637017.png" /></td> </tr></table>
+
$$
 +
f _ {k,l} ( \lambda )  = \
 +
 
 +
\frac{F _ {k,l} ( d \lambda ) }{d \lambda }
 +
.
 +
$$
  
 
In particular, if the relation
 
In particular, if the relation
  
<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/s08637018.png" /></td> </tr></table>
+
$$
 +
\sum _ {l = - \infty } ^  \infty 
 +
| B _ {k,l} ( t) |  < \infty ,\ \
 +
k , l = {1 \dots n } ,
 +
$$
 +
 
 +
holds for  $  X ( t) $,
 +
$  t = 0 , \pm  1 \dots $
 +
where
 +
 
 +
$$
 +
B ( t)  = \{ B _ {k,l} ( t) \} _ {k,l=} 1  ^ {n}  = \
 +
\{ {\mathsf E} X _ {k} ( t + s )
 +
\overline{ {X _ {l} ( s) }}\; \} _ {k,l=} 1  ^ {n}
 +
$$
  
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
+
is the covariance function of  $  X ( t) $,  
 +
then  $  X ( t) $
 +
has a spectral density and
  
<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/s08637021.png" /></td> </tr></table>
+
$$
 +
f _ {k,l} ( \lambda )  = ( 2 \pi )  ^ {-} 1
 +
\sum _ {t = - \infty } ^  \infty 
 +
B _ {k,l} ( t)  \mathop{\rm exp} \{ - i \lambda t \} ,
 +
$$
  
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
+
$$
 +
- \infty  < \lambda  <  \infty ,\  k , = {1 \dots n } .
 +
$$
  
<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/s08637024.png" /></td> </tr></table>
+
The situation is similar in the case of processes  $  X ( t) $
 +
in continuous time  $  t $.
 +
The spectral density  $  f ( \lambda ) $
 +
is sometimes called the second-order spectral density, in contrast to higher spectral densities (see [[Spectral semi-invariant|Spectral semi-invariant]]).
  
<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>
+
A homogeneous  $  n $-
 +
dimensional random field  $  X ( t _ {1} \dots t _ {n} ) $
 +
has a spectral density  $  f ( \lambda _ {1} \dots \lambda _ {n} ) $
 +
if its [[Spectral resolution|spectral resolution]]  $  F ( \lambda _ {1} \dots \lambda _ {n} ) $
 +
possesses the property that its mixed derivative  $  \partial  ^ {n} F / \partial  \lambda _ {1} \dots \partial  \lambda _ {n} $
 +
exists almost-everywhere, and then
  
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]]).
+
$$
 +
f ( \lambda _ {1} \dots \lambda _ {n} ) = \
  
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
+
\frac{\partial  ^ {n} F }{\partial  \lambda _ {1} \dots \partial  \lambda _ {n} }
  
<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/s08637034.png" /></td> </tr></table>
+
$$
  
 
and
 
and
  
<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/s08637035.png" /></td> </tr></table>
+
$$
 +
F ( \lambda _ {1} \dots \lambda _ {n} )  = \
 +
\int\limits _ { \lambda _ {01} } ^ { {\lambda _ 1 } } \dots
 +
\int\limits _ { \lambda _ {0n} } ^ { {\lambda _ n } }
 +
f ( \mu _ {1} \dots \mu _ {n} ) \
 +
d \mu _ {1} \dots d \mu _ {n} + \textrm{ const } .
 +
$$
  
 
====References====
 
====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>
 
<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>

Revision as of 08:22, 6 June 2020


of a stationary stochastic process or of a homogeneous random field in $ n $- 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

$$ X ( t) = \{ X _ {k} ( t) \} _ {k=} 1 ^ {n} $$

be an $ n $- dimensional stationary stochastic process, and let

$$ X ( t) = \int\limits e ^ {i t \lambda } \Phi ( d \lambda ) ,\ \Phi = \ \{ \Phi _ {k} \} _ {k=} 1 ^ {n} $$

be its spectral representation ( $ \Phi _ {k} $ is the spectral measure corresponding to the $ k $- th component $ X _ {k} ( t) $ of the multi-dimensional stochastic process $ X ( t) $). The range of integration is $ - \pi \leq \lambda \leq \pi $ in the case of discrete time $ t $, and $ - \infty < \lambda < + \infty $ in the case of continuous time $ t $. The process $ X ( t) $ has a spectral density

$$ f ( \lambda ) = \{ f _ {k,l} ( \lambda ) \} _ {k,l=} 1 ^ {n} , $$

if all the elements

$$ F _ {k,l} ( \Delta ) = {\mathsf E} \Phi _ {k} ( \Delta ) \overline{ {\Phi _ {l} ( \Delta ) }}\; ,\ \ k , l = {1 \dots n } , $$

of the spectral measure $ F = \{ F _ {k,l} \} _ {k,l=} 1 ^ {n} $ are absolutely continuous and if

$$ f _ {k,l} ( \lambda ) = \ \frac{F _ {k,l} ( d \lambda ) }{d \lambda } . $$

In particular, if the relation

$$ \sum _ {l = - \infty } ^ \infty | B _ {k,l} ( t) | < \infty ,\ \ k , l = {1 \dots n } , $$

holds for $ X ( t) $, $ t = 0 , \pm 1 \dots $ where

$$ B ( t) = \{ B _ {k,l} ( t) \} _ {k,l=} 1 ^ {n} = \ \{ {\mathsf E} X _ {k} ( t + s ) \overline{ {X _ {l} ( s) }}\; \} _ {k,l=} 1 ^ {n} $$

is the covariance function of $ X ( t) $, then $ X ( t) $ has a spectral density and

$$ f _ {k,l} ( \lambda ) = ( 2 \pi ) ^ {-} 1 \sum _ {t = - \infty } ^ \infty B _ {k,l} ( t) \mathop{\rm exp} \{ - i \lambda t \} , $$

$$ - \infty < \lambda < \infty ,\ k , l = {1 \dots n } . $$

The situation is similar in the case of processes $ X ( t) $ in continuous time $ t $. The spectral density $ f ( \lambda ) $ is sometimes called the second-order spectral density, in contrast to higher spectral densities (see Spectral semi-invariant).

A homogeneous $ n $- dimensional random field $ X ( t _ {1} \dots t _ {n} ) $ has a spectral density $ f ( \lambda _ {1} \dots \lambda _ {n} ) $ if its spectral resolution $ F ( \lambda _ {1} \dots \lambda _ {n} ) $ possesses the property that its mixed derivative $ \partial ^ {n} F / \partial \lambda _ {1} \dots \partial \lambda _ {n} $ exists almost-everywhere, and then

$$ f ( \lambda _ {1} \dots \lambda _ {n} ) = \ \frac{\partial ^ {n} F }{\partial \lambda _ {1} \dots \partial \lambda _ {n} } $$

and

$$ F ( \lambda _ {1} \dots \lambda _ {n} ) = \ \int\limits _ { \lambda _ {01} } ^ { {\lambda _ 1 } } \dots \int\limits _ { \lambda _ {0n} } ^ { {\lambda _ n } } f ( \mu _ {1} \dots \mu _ {n} ) \ d \mu _ {1} \dots d \mu _ {n} + \textrm{ const } . $$

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)
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