Spectral semi-invariant

spectral cumulant

One of the characteristics of a stationary stochastic process. Let , , be a real stationary stochastic process for which . The semi-invariants (cf. Semi-invariant) of this process,

are connected with the moments

by the relations

where

and the summation is over all partitions of into disjoint subsets . It is said that if, for all , there is a complex measure of bounded variation on such that for all ,

A measure , defined on a system of Borel sets, is called a spectral semi-invariant if, for all ,

The measure exists and has bounded variation if . In the case of a stationary process , the semi-invariants are invariant under translation:

and the spectral measures and are concentrated on the manifold . If the measure is absolutely continuous with respect to Lebesgue measure on this manifold, then there is a spectral density of order , defined by the equations

for all . In the case of discrete time one must replace in all formulas above by the -dimensional cube , .

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)