# 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 $- \pi \leq \lambda _ {i} \leq \pi$, $1 \leq i \leq k$.

How to Cite This Entry:
Spectral semi-invariant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spectral_semi-invariant&oldid=51963
This article was adapted from an original article by I.G. Zhurbenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article