Semi-invariant

A numerical characteristic of random variables related to the concept of a moment of higher order. If $ \xi = ( \xi _ {1} \dots \xi _ {k} ) $ is a random vector, $ \phi _ \xi ( t) = {\mathsf E} e ^ {i ( t, \xi ) } $ is its characteristic function, $ t = ( t _ {1} \dots t _ {k} ) $, $ t _ {i} \in \mathbf R $,

$$ ( t, \xi ) = \ \sum _ {i = 1 } ^ { k } t _ {i} \xi _ {i} , $$

and if for some $ n \geq 1 $ the moments $ {\mathsf E} | \xi _ {i} | ^ {n} < \infty $, $ i = 1 \dots k $, then the (mixed) moments

$$ m _ \xi ^ {( \nu _ {1} \dots \nu _ {k} ) } = \ {\mathsf E} \xi _ {1} ^ {\nu _ {1} } {} \dots \xi _ {k} ^ {\nu _ {k} } $$

exist for all non-negative integers $ \nu _ {1} \dots \nu _ {k} $ such that $ \nu _ {1} + \dots + \nu _ {k} \leq n $. Under these conditions,

$$ \phi _ \xi ( t) = \ \sum _ {\nu _ {1} + \dots + \nu _ {k} \leq n } \frac{i ^ {\nu _ {1} + \dots + \nu _ {k} } }{\nu _ {1} ! \dots \nu _ {k} ! } m _ \xi ^ {( \nu _ {1} \dots \nu _ {k} ) } \times $$

$$ \times t _ {1} ^ {\nu _ {1} } \dots t _ {k} ^ {\nu _ {k} } + o (| t | ^ {n} ), $$

where $ | t | = | t _ {1} | + \dots + | t _ {k} | $, and for sufficiently small $ | t | $ the principal value of $ \mathop{\rm ln} \phi _ \xi ( t) $ can be represented by Taylor's formula as

$$ \mathop{\rm ln} \phi _ \xi ( t) = \ \sum _ {\nu _ {1} + \dots + \nu _ {k} \leq n } \frac{i ^ {\nu _ {1} + \dots + \nu _ {k} } }{\nu _ {1} ! \dots \nu _ {k} ! } s _ \xi ^ {( \nu _ {1} \dots \nu _ {k} ) } \times $$

$$ \times t _ {1} ^ {\nu _ {1} } \dots t _ {k} ^ {\nu _ {k} } + o (| t | ^ {n} ), $$

where the coefficients $ s _ \xi ^ {( \nu _ {1} \dots \nu _ {k} ) } $ are called the (mixed) semi-invariants, or cumulants, of order $ \nu = ( \nu _ {1} \dots \nu _ {k} ) $ of the vector $ \xi = ( \xi _ {1} \dots \xi _ {k} ) $. For independent random vectors $ \xi = ( \xi _ {1} \dots \xi _ {k} ) $ and $ \eta = ( \eta _ {1} \dots \eta _ {k} ) $,

$$ s _ {\xi + \eta } ^ {( \nu _ {1} \dots \nu _ {k} ) } = \ s _ \xi ^ {( \nu _ {1} \dots \nu _ {k} ) } + s _ \eta ^ {( \nu _ {1} \dots \nu _ {k} ) } , $$

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:

$$ m _ \xi ^ {( \nu ) } = \ \sum ^ {*} { \frac{1}{q!} } \frac{\nu ! }{\lambda ^ {(} 1) ! \dots \lambda ^ {(} q) ! } \prod _ {p = 1 } ^ { q } s _ \xi ^ {( \lambda ^ {(} p) ) } , $$

$$ s _ \xi ^ {( \nu ) } = \sum ^ {*} \frac{(- 1) ^ {q - 1 } }{q } \frac{\nu ! }{\lambda ^ {(} 1) ! \dots \lambda ^ {(} q) ! } \prod _ {p = 1 } ^ { q } m _ \xi ^ {( \lambda ^ {(} p) ) } , $$

where $ \sum ^ {*} $ denotes summation over all ordered sets of non-negative integer vectors $ \lambda ^ {(} p) $, $ | \lambda ^ {(} p) | > 0 $, with as sum the vector $ \nu $. (Here $ v! $ is defined as $ v ! = v _ {1} ! \dots v _ {k} ! $, and similarly for the $ \lambda ^ {(} p) ! $.) In particular, if $ \xi $ is a random variable $ ( k = 1) $, $ m _ {n} = m _ \xi ^ {(} n) = {\mathsf E} \xi ^ {n} $, and $ s _ {n} = s _ \xi ^ {(} n) $, then

$$ m _ {1} = s _ {1} , $$

$$ m _ {2} = s _ {2} + s _ {1} ^ {2} , $$

$$ m _ {3} = s _ {3} + 3s _ {1} s _ {2} + s _ {1} ^ {3} , $$

$$ m _ {4} = s _ {4} + 3s _ {2} ^ {2} + 4s _ {1} s _ {3} + 6s _ {1} ^ {2} s _ {2} + s _ {1} ^ {4} , $$

and

$$ s _ {1} = m _ {1} (= {\mathsf E} \xi ), $$

$$ s _ {2} = m _ {2} - m _ {1} ^ {2} (= {\mathsf D} \xi ), $$

$$ s _ {3} = m _ {3} - 3m _ {1} m _ {2} + 2m _ {1} ^ {3} , $$

$$ s _ {4} = m _ {4} - 3m _ {2} ^ {2} - 4m _ {1} m _ {3} + 12m _ {1} ^ {2} m _ {2} - 6m _ {1} ^ {4} . $$

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