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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



[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)



[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. A.N. Shiryaev (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098