Let $(\kappa_i)_{i\ge1}$ be a sequence of constants. What is the sufficient condition for the existence of a unique distribution $X$ with these cumulants?
My guess is that, just like moments, it suffices to have the cumulant generating function, $$ \kappa(t) = \sum_{i \ge 1} \frac{\kappa_i}{i!} t^i $$ exists within a neighborhood of $0$, i.e., $\kappa(t) < \infty$ for $|t| < h$ where $h>0$. This in turn requires $\kappa(t)$ has a radius of convergence greater than $0$, i.e., $$ \limsup_{i \to \infty} \left(\frac{\kappa_i}{i!}\right)^{1/i} < \infty. $$ Is this a correct sufficient condition?
Moreover, assume that $(\kappa_i)_{i \ge 1}$ uniquely determines a distribution $X$. If we have a sequence of random variables $X_n$, such that $\kappa_i(X_n) \to \kappa_i$ for all $i \ge 0$, can we conclude that $X_n$ converges to $X$ in distribution?
Textbooks usually covers these two questions in terms of moments. I'd appreciate it if anyone can give some references regarding cumulants.