The normal/Gaussian distribution is fully characterized by its first and second statistical moments, $\mu$ and $\sigma$, so we can write the distribution only as a function of these two parameters; $F(\mu,\sigma)$ since the $n^{\text{th}}$ statistical moments vanish
My question is an attempt to generalize this notion and I haven't found much answering this (maybe I'm not phrasing it in the proper terminology). Is there any probability distribution that is fully characterized by its $n^{\text{th}}$ statistical moments?
(i.e. its $(n+1)^{\text{st}}$ statistical moment vanishes and its strictly a function $F(s_1,s_2,...,s_n)$ where $s_k$ is the $k^{\text{th}}$ statistical moment)