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