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) |
How to Cite This Entry:
Spectral semi-invariant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spectral_semi-invariant&oldid=48762
This article was adapted from an original article by I.G. Zhurbenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article