Suppose that $X_1,X_2,...,X_n$ are independent and identically distributed random variables. Known that for any unit vector $\mathbf q=[q_1,...,q_n]$ (i.e. $q_1^2+...+q_n^2=1$), $q_1X_1+...+q_nX_n$ and $X_1$ have the same distribution, then is it sufficient to derive that $X_1$ follows normal distribution with mean $0$?
As for the case where all positive integer moments of $X_1$ are finite, I proved the conclusion by calculating all positive integer moments of $X_1$ (by simplifying $\forall r=1,2,3,...,\mathbb E(q_1X_1+...+q_nX_n)^r=\mathbb EX_1^r$) and solving the Hamburger moment problem (see https://en.wikipedia.org/wiki/Hamburger_moment_problem, actually I only need to test that all positive integer moments of $X_1$ equal to those of a certain random variable following normal distribution with mean $0$, and prove the uniqueness of solutions of this Hamburger moment problem). However, is it possible that any high-order moment of $X_1$ is infinite (for example, fourth-order moment)?