The cumulant-generating function

Shmueli, U.; Wilson, A. J. C.

doi:10.1107/97809553602060000554

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 2.1, pp. 193-194 | 1 | 2 |

Section 2.1.4.2. The cumulant-generating function

U. Shmueli^a ^* and A. J. C. Wilson^b ^‡

^a School of Chemistry, Tel Aviv University, Tel Aviv 69 978, Israel, and ^bSt John's College, Cambridge, England
Correspondence e-mail: ushmueli@post.tau.ac.il

2.1.4.2. The cumulant-generating function

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A function that is often more useful than the characteristic function is its logarithm, the cumulant-generating function: $[K(t) = \log C(t) = k_{1}+{{k_{2}(it)^{2}}\over{2!}}+{{k_{3}(it)^{3}}\over{3!}}+ \ldots, \eqno(2.1.4.13)]$ where the k's are called the cumulants and may be regarded as being defined by the equation. They can be evaluated in terms of the moments by combining the series (2.1.4.10) for [C(t)] with the ordinary series for the logarithm and equating the coefficients of $[t^{r}]$ . In most cases the process as described is tedious, but it can be shortened by use of a general method [Stuart & Ord (1994), Section 3.14, pp. 87–88; Exercise 3.19, p. 119]. Obviously, the cumulants exist only if the moments exist. The first few relations are $[\eqalignno{k_{0} & = 0 \cr k_{1} & = m_{1}' \cr k_{2} & = m_{2} = m_{2}' - (m_{1}')^{2} &(2.1.4.14)\cr k_{3} & = m_{3} = m_{3}' - 3m_{2}'m_{1}' + 2(m_{1}')^{2} \cr k_{4} & = m_{4} - 3(m_{2})^{2} \cr & = m_{4}'-m_{3}'m_{1}'-3(m_{2}')^{2}+12m_{2}'(m_{1})^{2}-6(m_{1}')^{4}. }]$ Such expressions and their converses up to $[k_{10}]$ are given by Stuart & Ord (1994, pp. 88–91). Since all the cumulants except $[k_{1}]$ can be expressed in terms of the central moments only (i.e., those unprimed), only $[k_{1}]$ is changed by a change of the origin. Because of this property, they are sometimes called the semi-invariants (or seminvariants) of the distribution. Since addition of random variables is equivalent to the multiplication of their characteristic functions [equation (2.1.4.8)] and multiplication of functions is equivalent to the addition of their logarithms, each cumulant of the distribution of the sum of a number of random variables is equal to the sum of the cumulants of the distribution functions of the individual variables – hence the name cumulants. Although the cumulants (except $[k_{1}]$ ) are independent of a change of origin, they are not independent of a change of scale. As for the moments, a change of scale simply multiplies them by a power of the scale factor; if [y = x/b] $[(k_{y})_{r} = (k_{x})_{r}/b^{r}. \eqno(2.1.4.15)]$ The cumulants of the normal distribution are particularly simple. From equation (2.1.4.5), the cumulant-generating function of a normal distribution is $[\eqalignno{K(t) & = imt-{{\sigma^{2}t^{2}}/{2}} &(2.1.4.16) \cr k_{1} & = m &(2.1.4.17) \cr k_{2} & = \sigma^{2}, &(2.1.4.18)}%2.1.4.18]$ all cumulants with $[r \;\gt\; 2]$ are identically zero.