Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

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

Section The cumulant-generating function

U. Shmuelia* and A. J. C. Wilsonb

aSchool of Chemistry, Tel Aviv University, Tel Aviv 69 978, Israel, and bSt John's College, Cambridge, England
Correspondence e-mail: 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(] 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 ([link] 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[link]), 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} &(\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[link], 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 ([link]] 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(] The cumulants of the normal distribution are particularly simple. From equation ([link], the cumulant-generating function of a normal distribution is [\eqalignno{K(t) & = imt-{{\sigma^{2}t^{2}}/{2}} &( \cr k_{1} & = m &( \cr k_{2} & = \sigma^{2}, &(}%] all cumulants with [r \;\gt\; 2] are identically zero.


First citationStuart, A. & Ord, K. (1994). Kendall's advanced theory of statistics. Vol. 1. Distribution theory, 6th ed. London: Edward Arnold.Google Scholar

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