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m (fixing subscripts and superscripts)
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$$  
 
$$  
S  ^ {(} n) ( t _ {1} \dots t _ {n} )  = \  
+
S  ^ {( n)} ( t _ {1} \dots t _ {n} )  = \  
 
$$
 
$$
  
 
$$  
 
$$  
 
\left . =   
 
\left . =   
\frac{i  ^ {-} n \partial  ^ {n} }{\partial  u _ {1} \dots
+
\frac{i  ^ {- n} \partial  ^ {n} }{\partial  u _ {1} \dots
 
\partial  u _ {n} }
 
\partial  u _ {n} }
 
   \mathop{\rm log}  {\mathsf E} e ^ {i ( u _ {1} X
 
   \mathop{\rm log}  {\mathsf E} e ^ {i ( u _ {1} X
Line 34: Line 34:
  
 
$$  
 
$$  
M  ^ {(} n) ( t _ {1} \dots t _ {n} )  = \  
+
M  ^ {( n)} ( t _ {1} \dots t _ {n} )  = \  
 
{\mathsf E} \{ X ( t _ {1} ) \dots X ( t _ {n} ) \}
 
{\mathsf E} \{ X ( t _ {1} ) \dots X ( t _ {n} ) \}
 
$$
 
$$
Line 41: Line 41:
  
 
$$  
 
$$  
S  ^ {(} n) ( I)  =  \sum _ {\cup _ {p=} 1  ^ {q} I _ {p} = I }
+
S  ^ {( n)} ( I)  =  \sum _ {\cup _ {p=} 1  ^ {q} I _ {p} = I }
( - 1 )  ^ {q-} 1
+
( - 1 )  ^ {q- 1}
( q - 1 ) ! \prod _ { p= } 1 ^ { q }  M  ^ {(} p) ( I _ {p} ) ,
+
( q - 1 ) ! \prod _ { p= 1} ^ { q }  M  ^ {( p)} ( I _ {p} ) ,
 
$$
 
$$
  
 
$$  
 
$$  
M  ^ {(} n) ( I)  =  \sum _ {\cup _ {p=} 1 ^ {q} I _ {p} =
+
M  ^ {( n)} ( I)  =  \sum _ {\cup _ {p= 1}  ^ {q} I _ {p} =
I }  \prod _ { p= } 1 ^ { q }  S  ^ {(} p) ( I _ {p} ) ,
+
I }  \prod _ { p= 1} ^ { q }  S  ^ {( p)} ( I _ {p} ) ,
 
$$
 
$$
  
Line 60: Line 60:
 
and the summation is over all partitions of  $  I $
 
and the summation is over all partitions of  $  I $
 
into disjoint subsets  $  I _ {p} $.  
 
into disjoint subsets  $  I _ {p} $.  
It is said that  $  X ( t) \in \Phi  ^ {(} n) $
+
It is said that  $  X ( t) \in \Phi  ^ {( n)} $
 
if, for all  $  1 \leq  k \leq  n $,  
 
if, for all  $  1 \leq  k \leq  n $,  
there is a complex measure of bounded variation  $  M  ^ {(} k) $
+
there is a complex measure of bounded variation  $  M  ^ {( k)} $
 
on  $  \mathbf R  ^ {k} $
 
on  $  \mathbf R  ^ {k} $
 
such that for all  $  t _ {1} \dots t _ {n} $,
 
such that for all  $  t _ {1} \dots t _ {n} $,
  
 
$$  
 
$$  
M  ^ {(} k) ( t _ {1} \dots t _ {k} )  = \  
+
M  ^ {( k)} ( t _ {1} \dots t _ {k} )  = \  
 
\int\limits _ {\mathbf R  ^ {k} } e ^ {i
 
\int\limits _ {\mathbf R  ^ {k} } e ^ {i
( t _ {1} \lambda _ {1} + \dots + t _ {k} \lambda _ {k} ) } M  ^ {(} k)
+
( t _ {1} \lambda _ {1} + \dots + t _ {k} \lambda _ {k} ) } M  ^ {( k)}
 
( d \lambda _ {1} \dots d \lambda _ {k} ) =
 
( d \lambda _ {1} \dots d \lambda _ {k} ) =
 
$$
 
$$
Line 75: Line 75:
 
$$  
 
$$  
 
= \  
 
= \  
\int\limits _ {\mathbf R  ^ {k} } e ^ {i ( t , \lambda ) } M  ^ {(} k) ( d \lambda ) .
+
\int\limits _ {\mathbf R  ^ {k} } e ^ {i ( t , \lambda ) } M  ^ {( k)} ( d \lambda ) .
 
$$
 
$$
  
Line 82: Line 82:
  
 
$$  
 
$$  
S  ^ {(} n) ( t _ {1} \dots t _ {n} )  = \  
+
S  ^ {( n)} ( t _ {1} \dots t _ {n} )  = \  
 
\int\limits _ {\mathbf R  ^ {n} }
 
\int\limits _ {\mathbf R  ^ {n} }
 
e ^ {i ( t , \lambda ) } F ^ { ( n) } ( d \lambda ) .
 
e ^ {i ( t , \lambda ) } F ^ { ( n) } ( d \lambda ) .
Line 88: Line 88:
  
 
The measure  $  F ^ { ( n) } $
 
The measure  $  F ^ { ( n) } $
exists and has bounded variation if  $  X ( t) \in \Phi  ^ {(} n) $.  
+
exists and has bounded variation if  $  X ( t) \in \Phi  ^ {( n)} $.  
 
In the case of a stationary process  $  X ( t) $,  
 
In the case of a stationary process  $  X ( t) $,  
the semi-invariants  $  S  ^ {(} n) ( t _ {1} \dots t _ {n} ) $
+
the semi-invariants  $  S  ^ {( n)} ( t _ {1} \dots t _ {n} ) $
 
are invariant under translation:
 
are invariant under translation:
  
 
$$  
 
$$  
S  ^ {(} n) ( t _ {1} + \tau \dots t _ {n} +
+
S  ^ {( n)} ( t _ {1} + \tau \dots t _ {n} +
\tau )  =  S  ^ {(} n) ( t _ {1} \dots t _ {n} ) ,
+
\tau )  =  S  ^ {( n)} ( t _ {1} \dots t _ {n} ) ,
 
$$
 
$$
  
 
and the spectral measures  $  F ^ { ( n) } $
 
and the spectral measures  $  F ^ { ( n) } $
and  $  M  ^ {(} n) $
+
and  $  M  ^ {( n)} $
 
are concentrated on the manifold  $  \lambda _ {1} + \dots + \lambda _ {n} = 0 $.  
 
are concentrated on the manifold  $  \lambda _ {1} + \dots + \lambda _ {n} = 0 $.  
 
If the measure  $  F ^ { ( n) } $
 
If the measure  $  F ^ { ( n) } $
is absolutely continuous with respect to Lebesgue measure on this manifold, then there is a [[Spectral density|spectral density]]  $  f _ {n} ( \lambda _ {1} \dots \lambda _ {n-} 1 ) $
+
is absolutely continuous with respect to Lebesgue measure on this manifold, then there is a [[Spectral density|spectral density]]  $  f _ {n} ( \lambda _ {1} \dots \lambda _ {n- 1} ) $
 
of order  $  n $,  
 
of order  $  n $,  
 
defined by the equations
 
defined by the equations
  
 
$$  
 
$$  
S  ^ {(} n) ( t _ {1} \dots t _ {n} )  = \  
+
S  ^ {( n)} ( t _ {1} \dots t _ {n} )  = \  
\int\limits _ {\mathbf R  ^ {n-} 1 } e ^ {
+
\int\limits _ {\mathbf R  ^ {n- 1}} e ^ {
 
i ( \lambda _ {1} ( t _ {2} - t _ {1} ) +
 
i ( \lambda _ {1} ( t _ {2} - t _ {1} ) +
 
{} \dots + \lambda _ {n-} 1 ( t _ {n} - t _ {1} ) ) } \times
 
{} \dots + \lambda _ {n-} 1 ( t _ {n} - t _ {1} ) ) } \times
Line 119: Line 119:
  
 
for all  $  t _ {1} \dots t _ {n} $.  
 
for all  $  t _ {1} \dots t _ {n} $.  
In the case of discrete time one must replace  $  \mathbf R  ^ {(} k) $
+
In the case of discrete time one must replace  $  \mathbf R  ^ {( k)} $
in all formulas above by the  $  k $-
+
in all formulas above by the  $  k $-dimensional cube  $  - \pi \leq  \lambda _ {i} \leq  \pi $,  
dimensional cube  $  - \pi \leq  \lambda _ {i} \leq  \pi $,  
 
 
$  1 \leq  i \leq  k $.
 
$  1 \leq  i \leq  k $.
  
 
====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">  V.P. Leonov,  "Some applications of higher semi-invariants to the theory of stationary stochastic processes" , Moscow  (1964)  (In 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">  V.P. Leonov,  "Some applications of higher semi-invariants to the theory of stationary stochastic processes" , Moscow  (1964)  (In Russian)</TD></TR></table>

Revision as of 02:04, 21 January 2022


spectral cumulant

One of the characteristics of a stationary stochastic process. Let $ X ( t) $, $ - \infty < t < \infty $, be a real stationary stochastic process for which $ {\mathsf E} | X ( t) | ^ {n} \leq C < \infty $. The semi-invariants (cf. Semi-invariant) of this process,

$$ S ^ {( n)} ( t _ {1} \dots t _ {n} ) = \ $$

$$ \left . = \frac{i ^ {- n} \partial ^ {n} }{\partial u _ {1} \dots \partial u _ {n} } \mathop{\rm log} {\mathsf E} e ^ {i ( u _ {1} X ( t _ {1} ) + \dots + u _ {n} X ( t _ {n} ) ) } \right | _ {u _ {1} = \dots = u _ {n} = 0 } , $$

are connected with the moments

$$ M ^ {( n)} ( t _ {1} \dots t _ {n} ) = \ {\mathsf E} \{ X ( t _ {1} ) \dots X ( t _ {n} ) \} $$

by the relations

$$ S ^ {( n)} ( I) = \sum _ {\cup _ {p=} 1 ^ {q} I _ {p} = I } ( - 1 ) ^ {q- 1} ( q - 1 ) ! \prod _ { p= 1} ^ { q } M ^ {( p)} ( I _ {p} ) , $$

$$ M ^ {( n)} ( I) = \sum _ {\cup _ {p= 1} ^ {q} I _ {p} = I } \prod _ { p= 1} ^ { q } S ^ {( p)} ( I _ {p} ) , $$

where

$$ I = ( t _ {1} \dots t _ {n} ) ,\ \ I _ {p} = ( t _ {i _ {1} } \dots t _ {i _ {p} } ) \subseteq I , $$

and the summation is over all partitions of $ I $ into disjoint subsets $ I _ {p} $. It is said that $ X ( t) \in \Phi ^ {( n)} $ if, for all $ 1 \leq k \leq n $, there is a complex measure of bounded variation $ M ^ {( k)} $ on $ \mathbf R ^ {k} $ such that for all $ t _ {1} \dots t _ {n} $,

$$ M ^ {( k)} ( t _ {1} \dots t _ {k} ) = \ \int\limits _ {\mathbf R ^ {k} } e ^ {i ( t _ {1} \lambda _ {1} + \dots + t _ {k} \lambda _ {k} ) } M ^ {( k)} ( d \lambda _ {1} \dots d \lambda _ {k} ) = $$

$$ = \ \int\limits _ {\mathbf R ^ {k} } e ^ {i ( t , \lambda ) } M ^ {( k)} ( d \lambda ) . $$

A measure $ F ^ { ( n) } $, defined on a system of Borel sets, is called a spectral semi-invariant if, for all $ t _ {1} \dots t _ {n} $,

$$ S ^ {( n)} ( t _ {1} \dots t _ {n} ) = \ \int\limits _ {\mathbf R ^ {n} } e ^ {i ( t , \lambda ) } F ^ { ( n) } ( d \lambda ) . $$

The measure $ F ^ { ( n) } $ exists and has bounded variation if $ X ( t) \in \Phi ^ {( n)} $. In the case of a stationary process $ X ( t) $, the semi-invariants $ S ^ {( n)} ( t _ {1} \dots t _ {n} ) $ are invariant under translation:

$$ S ^ {( n)} ( t _ {1} + \tau \dots t _ {n} + \tau ) = S ^ {( n)} ( t _ {1} \dots t _ {n} ) , $$

and the spectral measures $ F ^ { ( n) } $ and $ M ^ {( n)} $ are concentrated on the manifold $ \lambda _ {1} + \dots + \lambda _ {n} = 0 $. If the measure $ F ^ { ( n) } $ is absolutely continuous with respect to Lebesgue measure on this manifold, then there is a spectral density $ f _ {n} ( \lambda _ {1} \dots \lambda _ {n- 1} ) $ of order $ n $, defined by the equations

$$ S ^ {( n)} ( t _ {1} \dots t _ {n} ) = \ \int\limits _ {\mathbf R ^ {n- 1}} e ^ { i ( \lambda _ {1} ( t _ {2} - t _ {1} ) + {} \dots + \lambda _ {n-} 1 ( t _ {n} - t _ {1} ) ) } \times $$

$$ \times f _ {n} ( \lambda _ {1} \dots \lambda _ {n-} 1 ) d \lambda , $$

for all $ t _ {1} \dots t _ {n} $. In the case of discrete time one must replace $ \mathbf R ^ {( k)} $ in all formulas above by the $ k $-dimensional cube $ - \pi \leq \lambda _ {i} \leq \pi $, $ 1 \leq i \leq k $.

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] V.P. Leonov, "Some applications of higher semi-invariants to the theory of stationary stochastic processes" , Moscow (1964) (In Russian)
How to Cite This Entry:
Spectral semi-invariant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spectral_semi-invariant&oldid=51961
This article was adapted from an original article by I.G. Zhurbenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article