A numerical characteristic of random variables related to the concept of a moment of higher order. If
is a random vector,
is its characteristic function,
,
,
and if for some
the moments
,
, then the (mixed) moments
exist for all non-negative integers
such that
. Under these conditions,
where
, and for sufficiently small
the principal value of
can be represented by Taylor's formula as
where the coefficients
are called the (mixed) semi-invariants, or cumulants, of order
of the vector
. For independent random vectors
and
,
that is, the semi-invariant of a sum of independent random vectors is the sum of their semi-invariants. This is the reason for the term "semi-invariant" , which reflects the additive property of independent variables (but, in general, the property does not hold for dependent variables).
The following formulas, connecting moments and semi-invariants, hold:
where
denotes summation over all ordered sets of non-negative integer vectors
,
, with as sum the vector
. (Here
is defined as
, and similarly for the
.) In particular, if
is a random variable
,
, and
, then
and
References
[1] | V.P. Leonov, A.N. Shiryaev, "On a method of calculation of semi-invariants" Theory Probab. Appl. , 4 : 3 (1959) pp. 319–329 Teor. Veroyatnost. i Primen. , 4 : 3 (1959) pp. 342–355 |
[2] | A.N. Shiryaev, "Probability" , Springer (1984) (Translated from Russian) |
References
[a1] | A. Stuart, J.K. Ord, "Kendall's advanced theory of statistics" , Griffin (1987) |
[a2] | L. Schmetterer, "Introduction to mathematical statistics" , Springer (1974) pp. Chapt. 1, §42 (Translated from German) |
[a3] | A. Rényi, "Probability theory" , North-Holland (1970) pp. Chapt. 3, §15 |
How to Cite This Entry:
Semi-invariant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-invariant&oldid=48662
This article was adapted from an original article by A.N. Shiryaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article