# Proof for additivity of cumulants

If one does not define cumulants via the cumulant generating function (cgf), e.g. because the cgf does not exist, then an alternative way is to use the recusion \begin{align*} \kappa_n=\mu'_n-\sum_{m=1}^{n-1}{n-1 \choose m-1}\kappa_m \mu_{n-m}', \end{align*} where $\mu_i'$ denotes the $i$th uncentered moment.

For this definition, what is the best way to show that the cumulants are additive under an independence assumption? More precisely, how do we show that \begin{align*} \kappa_n(X+Y) = \kappa_n(X) + \kappa_n(Y), \end{align*} if $X$ and $Y$ are two independent random variables?