According to Wikipedia, the $n$th cumulant $\kappa_n$ is related to the central-moments $\theta_n$ by the following recurrence:

$$\kappa_n = \theta_n - \sum_{m=1}^{n-1} \binom{n-1}{m-1} \kappa_m \theta_{n-m}$$

which allows one to compute the cumulants sequentially, given the central-moments. The Wikipedia page offers no reference or proof of this formula. What's the proof of this formula?


The equation given by Wikipedia connects cumulants to moments (generally).

A proof of a formula connecting cumulants to central moments is found in A Recursive Formulation of the Old Problem of Obtaining Moments from Cumulants and Vice Versa

Letting $K(t)$ be the cumulant-generating function, and $M(t)$ the moment-generating function. The relationship between the two is \begin{equation} M(t)=\exp{\left[K(t)\right]} \end{equation} The proof follows by differentiation of this expression and noting that the $n$th derivative can be written as \begin{equation} D^n[M(t)]=\sum_{i=0}^{n-1}\binom{n-1}{i}D^{n-i}[K(t)]D^i[M(t)] \end{equation} Where $D^k$ denotes the $k$th derivative. Now setting $t=0$: \begin{equation} \theta_n=\sum_{i=0}^{n-1}\binom{n-1}{i}\kappa_{n-i}\theta_i\\ \theta_n=\kappa_n+\sum_{i=1}^{n-1}\binom{n-1}{i}\kappa_{n-i}\theta_i\\ \end{equation} Rewriting yields: \begin{equation} \kappa_n = \theta_n-\sum_{i=1}^{n-1}\binom{n-1}{i}\kappa_{n-i}\theta_i \end{equation} In terms of the central moments and cumulants.


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