Suppose that we have a sequence of random variable $X(n)$ and we know that each $X(n)$ has moments of all orders. The well-known `method of moments' tells us that if, for each $k \ge 1$, $\mathbb{E}(X(n)^k) \to \mathbb{E}(X^k)$ as $n\to\infty$ where $X$ is a normal distribution then $X(n)$ converges in distribution to $X$.
My question is as follows: is the same true for higher dimensional central limit theorems?
That is, if $X_1(n), \ldots, X_k(n)$ are sequences of $\mathbb{R}$ valued random variables and we know that for every $p_1,\ldots, p_k \in \mathbb{N}$ the moments $$ \mathbb{E}(X_1(n)^{p_1}\cdots X_k(n)^{p_k}) $$ converge as $n\to\infty$ to the corresponding moments of a $k$-dimensional normal distribution, then does $(X_1(n),\ldots, X_k(n))$ converge in distribution to a $k$-dimensional normal distribution?
Thanks is advance for any help!