Method of moments and limiting distribution

Question

Suppose that $X_1, X_2,\dots$ are random variables (taking positive real values, say) so that for all $k\ge 1$ $\lim_{n\to\infty}\mathbb E[X_n^k] = C_k$ for some $k$ where the $C_k$ are positive and grow at a reasonably tame rate (so that the moment generating function has a positive radius of convergence). Does this imply that there exists some limiting distribution for the $X_j$. In particular, can one conclude that there is some Borel measure $\mu$ so that for any measurable $A\subset\mathbb R$, we have that $$\lim_{n\to\infty}\mathbb{P}(X_n\in A) = \mu(A),$$ and if not, what other information is required? Typically, in applications of the method of moments, I find that typically, what $\mu$ ought to be tends to be known, so the problem reduces to the uniqueness of moment generating functions. This seems like something that ought to be standard, though I have been unable to find any reference that applies in this situation.

Answer 1

It is known that there exists a positive Borel measure $\mu$ on $\mathbb R$ satisfying $\mu(x^k)=C_k$ for all integers $k\geq 0$ if and only if for all $t_1,\ldots,t_n\in\mathbb C$ one has the inequality $$ \sum_{j,k=1}^n C_{j+k}t_j\overline{t_k}\geq 0. $$ This holds for your sequence $C_k$, since $$ \sum_{j,k=1}^n C_{j+k}t_j\overline{t_k}=\lim_{m\to\infty}\mathbb E\Bigl|\sum_{i=1}^n t_i X_m^k \Bigr|^2\geq 0. $$

If you are also given a suitable growth condition on $C_k$, then you will have uniqueness of $\mu$ as well - see here for more details.

Once you have uniqueness of $\mu$, then the convergence you seek follows from the argument given here.