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''of a real stochastic process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026540/c0265401.png" />''
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The function in the arguments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026540/c0265402.png" /> defined by
+
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 +
{{TEX|done}}
  
<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/c/c026/c026540/c0265403.png" /></td> </tr></table>
+
''of a real stochastic process  $  \{ {X ( t) } : {t \in T } \} $''
  
For the correlation function to be defined, it must be assumed that the process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026540/c0265404.png" /> has a finite second moment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026540/c0265405.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026540/c0265406.png" />. The parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026540/c0265407.png" /> varies here over some subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026540/c0265408.png" /> of the real line; it is usually interpreted as "time" , though an entirely analogous definition is possible for the correlation function of a stochastic field, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026540/c0265409.png" /> is a subset of a finite-dimensional space. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026540/c02654010.png" /> is a multivariate stochastic process (stochastic function), then its correlation function is defined to be the matrix-valued function
+
The function in the arguments  $ t, s \in T $
 +
defined by
  
<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/c/c026/c026540/c02654011.png" /></td> </tr></table>
+
$$
 +
B ( t, s)  = \
 +
{\mathsf E} [ X ( t) - {\mathsf E} X ( t)]
 +
[ X ( s) - {\mathsf E} X ( s)].
 +
$$
 +
 
 +
For the correlation function to be defined, it must be assumed that the process  $  X ( t) $
 +
has a finite second moment  $  {\mathsf E} X ( t)  ^ {2} $
 +
for all  $  t \in T $.
 +
The parameter  $  t $
 +
varies here over some subset  $  T $
 +
of the real line; it is usually interpreted as  "time" , though an entirely analogous definition is possible for the correlation function of a stochastic field, where  $  T $
 +
is a subset of a finite-dimensional space. If  $  \mathbf X ( t) = [ X _ {1} ( t) \dots X _ {n} ( t)] $
 +
is a multivariate stochastic process (stochastic function), then its correlation function is defined to be the matrix-valued function
 +
 
 +
$$
 +
B ( t, s)  = \
 +
\| B _ {ij} ( t, s) \| _ {i, j = 1 }  ^ {n} ,
 +
$$
  
 
where
 
where
  
<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/c/c026/c026540/c02654012.png" /></td> </tr></table>
+
$$
 +
B _ {ij} ( t, s)  = \
 +
{\mathsf E} [ X _ {i} ( t) - {\mathsf E} X _ {i} ( t) ]
 +
[ X _ {j} ( s) - {\mathsf E} X _ {j} ( s)]
 +
$$
  
is the joint correlation function of the processes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026540/c02654013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026540/c02654014.png" />.
+
is the joint correlation function of the processes $  X _ {i} ( t) $,  
 +
$  X _ {j} ( t) $.
  
The correlation function is an important characteristic of a stochastic process. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026540/c02654015.png" /> is a [[Gaussian process|Gaussian process]], then its correlation function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026540/c02654016.png" /> and its mean value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026540/c02654017.png" /> (i.e. its first and second moments) uniquely determine its finite-dimensional distributions; hence also the process as a whole. In the general case, the first two moments are known to be insufficient for a full description of a stochastic process. For example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026540/c02654018.png" /> is at one and the same time the correlation function of a stationary Gaussian Markov process the trajectories of which are continuous, and also the correlation function of the so-called telegraph signal, a stationary Markov point process taking the two values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026540/c02654019.png" />. However, the correlation function does determine several important properties of a process: the so-called second-order properties (i.e. properties expressed in terms of second moments). In view of this, and also because of their relative simplicity, correlation methods are frequently employed both in the theory of stochastic processes and in its statistical applications (see [[Correlogram|Correlogram]]).
+
The correlation function is an important characteristic of a stochastic process. If $  X ( t) $
 +
is a [[Gaussian process|Gaussian process]], then its correlation function $  B ( t, s) $
 +
and its mean value $  {\mathsf E} X ( t) $(
 +
i.e. its first and second moments) uniquely determine its finite-dimensional distributions; hence also the process as a whole. In the general case, the first two moments are known to be insufficient for a full description of a stochastic process. For example, $  B ( t, s) = e ^ {- a | t - s | } $
 +
is at one and the same time the correlation function of a stationary Gaussian Markov process the trajectories of which are continuous, and also the correlation function of the so-called telegraph signal, a stationary Markov point process taking the two values $  \pm  1 $.  
 +
However, the correlation function does determine several important properties of a process: the so-called second-order properties (i.e. properties expressed in terms of second moments). In view of this, and also because of their relative simplicity, correlation methods are frequently employed both in the theory of stochastic processes and in its statistical applications (see [[Correlogram|Correlogram]]).
  
The rate and nature of decrease of the correlations as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026540/c02654020.png" /> provides an idea of the ergodic properties of a process. Conditions relating to the rate of decrease of correlations, in some form or another, appear in limit theorems for stochastic processes. Local second-order properties, such as mean-square continuity and differentiability, provide a useful — though extremely crude — characteristic of the local behaviour of a process. The properties of the trajectories in terms of the correlation function have been investigated to a considerable degree in the Gaussian case (see [[Sample function|Sample function]]). One of the most complete branches of the theory of stochastic processes is the theory of linear extrapolation and filtration, which yields optimal linear algorithms for the prediction and approximation of stochastic processes; this theory is based on a knowledge of the correlation function.
+
The rate and nature of decrease of the correlations as $  | t - s | \rightarrow \infty $
 +
provides an idea of the ergodic properties of a process. Conditions relating to the rate of decrease of correlations, in some form or another, appear in limit theorems for stochastic processes. Local second-order properties, such as mean-square continuity and differentiability, provide a useful — though extremely crude — characteristic of the local behaviour of a process. The properties of the trajectories in terms of the correlation function have been investigated to a considerable degree in the Gaussian case (see [[Sample function|Sample function]]). One of the most complete branches of the theory of stochastic processes is the theory of linear extrapolation and filtration, which yields optimal linear algorithms for the prediction and approximation of stochastic processes; this theory is based on a knowledge of the correlation function.
  
 
A characteristic property of the correlation function is the fact that it is positive definite:
 
A characteristic property of the correlation function is the fact that it is positive definite:
  
<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/c/c026/c026540/c02654021.png" /></td> </tr></table>
+
$$
 +
\sum _ {i, j = 1 } ^ { n }
 +
c _ {i} \overline{c}\; _ {j} B ( t _ {i} , t _ {j} )
 +
\geq  0 ,
 +
$$
  
for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026540/c02654022.png" />, any complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026540/c02654023.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026540/c02654024.png" />. In the most important case of a stationary process in the broad sense, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026540/c02654025.png" /> depends (only) on the difference between the arguments: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026540/c02654026.png" />. The condition that it be positive definite then becomes
+
for any $  n $,  
 +
any complex $  c _ {1} \dots c _ {n} $
 +
and any $  t _ {1} \dots t _ {n} \in T $.  
 +
In the most important case of a stationary process in the broad sense, $  B ( t, s) $
 +
depends (only) on the difference between the arguments: $  B ( t, s) = R ( t - s) $.  
 +
The condition that it be positive definite then becomes
  
<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/c/c026/c026540/c02654027.png" /></td> </tr></table>
+
$$
 +
\sum _ {i, j = 1 } ^ { n }
 +
c _ {i} \overline{c}\; _ {j} R ( t _ {i} - t _ {j} )
 +
\geq  0.
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026540/c02654028.png" /> is also continuous at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026540/c02654029.png" /> (in other words, the process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026540/c02654030.png" /> is mean-square continuous), then
+
If $  R ( t) $
 +
is also continuous at $  t = 0 $(
 +
in other words, the process $  X ( t) $
 +
is mean-square continuous), then
  
<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/c/c026/c026540/c02654031.png" /></td> </tr></table>
+
$$
 +
R ( t)  = \int\limits
 +
e ^ {it \lambda }
 +
F ( d \lambda ),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026540/c02654032.png" /> is a positive finite measure; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026540/c02654033.png" /> runs over the entire real line if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026540/c02654034.png" /> (the case of  "continuous time" ), or over the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026540/c02654035.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026540/c02654036.png" /> (the case of  "discrete time" ). The measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026540/c02654037.png" /> is known as the spectral measure of the stochastic process. Thus, the correlation and spectral properties of a stationary stochastic process prove to be closely related; for example, the rate of decrease in correlations as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026540/c02654038.png" /> corresponds to the degree of smoothness of the spectral density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026540/c02654039.png" />, etc.
+
where $  F ( d \lambda ) $
 +
is a positive finite measure; here $  \lambda $
 +
runs over the entire real line if $  T = (- \infty , \infty ) $(
 +
the case of  "continuous time" ), or over the interval $  [- \pi , \pi ] $
 +
if $  T = \{ \dots, - 1, 0, 1 ,\dots \} $(
 +
the case of  "discrete time" ). The measure $  F ( d \lambda ) $
 +
is known as the spectral measure of the stochastic process. Thus, the correlation and spectral properties of a stationary stochastic process prove to be closely related; for example, the rate of decrease in correlations as $  t \rightarrow \infty $
 +
corresponds to the degree of smoothness of the spectral density $  f ( \lambda ) = F ( d \lambda )/d \lambda $,  
 +
etc.
  
In statistical mechanics, the term is also used for the joint probability density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026540/c02654040.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026540/c02654041.png" /> distinct particles of the system under consideration placed at points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026540/c02654042.png" />; the totality of these functions uniquely determines the corresponding discrete stochastic field.
+
In statistical mechanics, the term is also used for the joint probability density $  \rho ( x _ {1} \dots x _ {m} ) $
 +
of $  m $
 +
distinct particles of the system under consideration placed at points $  x _ {1} \dots x _ {m} $;  
 +
the totality of these functions uniquely determines the corresponding discrete stochastic field.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.L. Doob,  "Stochastic processes" , Chapman &amp; Hall  (1953)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Loève,  "Probability theory" , Princeton Univ. Press  (1963)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.I. Gikhman,  A.V. Skorokhod,  "Introduction to the theory of stochastic processes" , Saunders  (1969)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.L. Doob,  "Stochastic processes" , Chapman &amp; Hall  (1953)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Loève,  "Probability theory" , Princeton Univ. Press  (1963)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.I. Gikhman,  A.V. Skorokhod,  "Introduction to the theory of stochastic processes" , Saunders  (1969)  (Translated from Russian)</TD></TR></table>

Latest revision as of 17:31, 5 June 2020


of a real stochastic process $ \{ {X ( t) } : {t \in T } \} $

The function in the arguments $ t, s \in T $ defined by

$$ B ( t, s) = \ {\mathsf E} [ X ( t) - {\mathsf E} X ( t)] [ X ( s) - {\mathsf E} X ( s)]. $$

For the correlation function to be defined, it must be assumed that the process $ X ( t) $ has a finite second moment $ {\mathsf E} X ( t) ^ {2} $ for all $ t \in T $. The parameter $ t $ varies here over some subset $ T $ of the real line; it is usually interpreted as "time" , though an entirely analogous definition is possible for the correlation function of a stochastic field, where $ T $ is a subset of a finite-dimensional space. If $ \mathbf X ( t) = [ X _ {1} ( t) \dots X _ {n} ( t)] $ is a multivariate stochastic process (stochastic function), then its correlation function is defined to be the matrix-valued function

$$ B ( t, s) = \ \| B _ {ij} ( t, s) \| _ {i, j = 1 } ^ {n} , $$

where

$$ B _ {ij} ( t, s) = \ {\mathsf E} [ X _ {i} ( t) - {\mathsf E} X _ {i} ( t) ] [ X _ {j} ( s) - {\mathsf E} X _ {j} ( s)] $$

is the joint correlation function of the processes $ X _ {i} ( t) $, $ X _ {j} ( t) $.

The correlation function is an important characteristic of a stochastic process. If $ X ( t) $ is a Gaussian process, then its correlation function $ B ( t, s) $ and its mean value $ {\mathsf E} X ( t) $( i.e. its first and second moments) uniquely determine its finite-dimensional distributions; hence also the process as a whole. In the general case, the first two moments are known to be insufficient for a full description of a stochastic process. For example, $ B ( t, s) = e ^ {- a | t - s | } $ is at one and the same time the correlation function of a stationary Gaussian Markov process the trajectories of which are continuous, and also the correlation function of the so-called telegraph signal, a stationary Markov point process taking the two values $ \pm 1 $. However, the correlation function does determine several important properties of a process: the so-called second-order properties (i.e. properties expressed in terms of second moments). In view of this, and also because of their relative simplicity, correlation methods are frequently employed both in the theory of stochastic processes and in its statistical applications (see Correlogram).

The rate and nature of decrease of the correlations as $ | t - s | \rightarrow \infty $ provides an idea of the ergodic properties of a process. Conditions relating to the rate of decrease of correlations, in some form or another, appear in limit theorems for stochastic processes. Local second-order properties, such as mean-square continuity and differentiability, provide a useful — though extremely crude — characteristic of the local behaviour of a process. The properties of the trajectories in terms of the correlation function have been investigated to a considerable degree in the Gaussian case (see Sample function). One of the most complete branches of the theory of stochastic processes is the theory of linear extrapolation and filtration, which yields optimal linear algorithms for the prediction and approximation of stochastic processes; this theory is based on a knowledge of the correlation function.

A characteristic property of the correlation function is the fact that it is positive definite:

$$ \sum _ {i, j = 1 } ^ { n } c _ {i} \overline{c}\; _ {j} B ( t _ {i} , t _ {j} ) \geq 0 , $$

for any $ n $, any complex $ c _ {1} \dots c _ {n} $ and any $ t _ {1} \dots t _ {n} \in T $. In the most important case of a stationary process in the broad sense, $ B ( t, s) $ depends (only) on the difference between the arguments: $ B ( t, s) = R ( t - s) $. The condition that it be positive definite then becomes

$$ \sum _ {i, j = 1 } ^ { n } c _ {i} \overline{c}\; _ {j} R ( t _ {i} - t _ {j} ) \geq 0. $$

If $ R ( t) $ is also continuous at $ t = 0 $( in other words, the process $ X ( t) $ is mean-square continuous), then

$$ R ( t) = \int\limits e ^ {it \lambda } F ( d \lambda ), $$

where $ F ( d \lambda ) $ is a positive finite measure; here $ \lambda $ runs over the entire real line if $ T = (- \infty , \infty ) $( the case of "continuous time" ), or over the interval $ [- \pi , \pi ] $ if $ T = \{ \dots, - 1, 0, 1 ,\dots \} $( the case of "discrete time" ). The measure $ F ( d \lambda ) $ is known as the spectral measure of the stochastic process. Thus, the correlation and spectral properties of a stationary stochastic process prove to be closely related; for example, the rate of decrease in correlations as $ t \rightarrow \infty $ corresponds to the degree of smoothness of the spectral density $ f ( \lambda ) = F ( d \lambda )/d \lambda $, etc.

In statistical mechanics, the term is also used for the joint probability density $ \rho ( x _ {1} \dots x _ {m} ) $ of $ m $ distinct particles of the system under consideration placed at points $ x _ {1} \dots x _ {m} $; the totality of these functions uniquely determines the corresponding discrete stochastic field.

References

[1] J.L. Doob, "Stochastic processes" , Chapman & Hall (1953)
[2] M. Loève, "Probability theory" , Princeton Univ. Press (1963)
[3] I.I. Gikhman, A.V. Skorokhod, "Introduction to the theory of stochastic processes" , Saunders (1969) (Translated from Russian)
How to Cite This Entry:
Correlation function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Correlation_function&oldid=46523
This article was adapted from an original article by A.S. Kholevo (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article