If the moments of a sequence of distributions converge, do they represent a probability? Assume ${\bf x} \in \mathbb{R}^n$ denotes a real-valued and bounded random variable. Then, the moments of $\bf x$ uniquely define its distribution. 
Assume we have not only one distribution, but a sequence indexed by $n\in\mathbb{N}$. Let $M_{k,n}$ denote the $k$-th mixed moment of the $n$-th distribution.
If the limit 
$~\lim_{n\to\infty} M_{k,n}=M_{k}$
exists for all $k$, can we conclude that the $M_{k}$ are the moments of a distribution?
 A: Here is the 1-d version of proposition that the question is looking for, which I believe to be correct (but idk for sure):
Proposition: Suppose that $X_n \in \mathbb{R}$ is a sequence of random variables such that: 


*

*$|X_n| \leq N$ 

*$\lim E[X_n^k] = m_k \in \mathbb{R}$ for all $k = 1, \ldots, \infty$
Then there is a random variable $X$ with $E[X^k] = m_k$.
Proof:
 First, we observe that the sequence of random variables $X_n$ is tight, using: Tightness of a sequence of random variables with bounded mean and variance . 
Since you are assuming the random variables are universally bounded, they are tight.
By Prokohorov's theorem (in particular the corollary for measures in $\mathbb{R}^m$), tightness implies that there is a subsequence $X_{n_k}$ which converges in distribution to some random variable $X$. We can pass to this subsequence, since this does not effect the two hypothesis.
Now, for any $k \geq 1$, let $f_k(x)$ be any function that agrees with $x^k$ on $[-N, N]$ and which is bounded and continuous. For instance, $f(x)$ might linearly interpolate so that it is zero on $(\infty, -N - 1) \cup (N + 1, \infty)$. 
Since we have $X_n \to X$ weakly, we have from the Portmanteau theorem that $E [ f_k(X_n) ] \to E[f(X)]$. But since $|X_n| \leq N$ and $|X| \leq N$ a.s. (this is a consequence of weak convergence, e.g. using the CDF), we have that $f_k(x) = x^k$ on $[-N, N]$, which means that $E[X_n^k] \to E[X^k]$. By the second assumption, this implies that $E[X^k] = m_k$, which is the desired conclusion. QED
Hope this helps! 
I think it should be relatively straightforward to adapt to $\mathbb{R}^n$. 
We don't have to worry about tightness because of your assumption about the universal boundedness:  You just want to make sure you that your assumption about moments again implies tightness -- I'm not completely sure what mixed moments you are controlling, but if you get control over the first two moments of $||X_n||_2$ (or $L_p$ norms for some $p$), you can prove that the $||X_n||_2$ are tight, which implies that $X_n$ are. 
(I'm not sure that controlling moments like $E[X_1^k X_2^k]$ is enough; e.g. its possible that $X_1$ blows up a lot for small $X_2$, and maybe mass can escape to infinity that way? But if you know that all moments of the form $E[X_i^k]$ have limits I think you are in good shape.) 
