Spectral semi-invariant

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)