What sequences of real-valued random variables $X_1,X_2,X_3,\ldots$ exist for which for all $n$ and all $k$ $$ \operatorname{cum}_k (X_1+\cdots+X_n) = \operatorname{cum}_k(X_1)+\cdots + \operatorname{cum}_k(X_n) $$ where $\operatorname{cum}_k$ is the $k$th cumulant functional, and $X_1,X_2,X_3,\ldots$ are not independent?
$$ \S $$
PS: The most usual definitions of "cumulant" that one comes across don't make clear any motivation behind the concept, so here is one that does: The $k$th cumulant $\operatorname{cum}_k(X)$ of (the probability distribution of) a random variable $X$ is the value of a certain polynomial in the first $k$ moments $\operatorname{E}(X^\ell),\ \ell=1,\ldots,k$. The polynomial is the unique one that is so chosen that
- $\operatorname{cum}_k$ is shift-invariant for $k\ge 2$, i.e. $\operatorname{cum}_k(X+\text{constant}) = \operatorname{cum}_k(X)$ (and for $k=1$ it is shift-equivariant, i.e. $\operatorname{cum}_k(X+\text{constant}) = \operatorname{cum}_k(X)+\text{same constant}$); and
- $\operatorname{cum}_k$ is homogeneous of degree $k$; and
- $\operatorname{cum}_k$ is "cumulative", i.e. for independent random variables $X_1,X_2,X_3,\ldots$ one as for all $n$, $\operatorname{cum}_k(X_1+\cdots+X_n) = \operatorname{cum}_k(X_1) + \cdots + \operatorname{cum}_k(X_n)$.
(For example, the $4$th cumulant is the $4$th central moment minus $3$ times the square of the second central moment.)
(I suppose this characterization doesn't really explain why $k=1$ should be an exception in the first bulleted item above, where perhaps the more usual characterizations do.)
If all of the cumulants are additive, that means the cumulant generating function is additive, meaning that $\log \textbf{E} e^{-itX_1} + \log \textbf{E} e^{-it X_2} = \log \textbf{E} e^{-it(X_1+X_2)}$.
Now your question reduces to A criterion for independence based on Characteristic function, and it turns out the variables do not have to be independent.