# Convergence of moments implies convergence to normal distribution

I have a sequence $\{X_n\}$ of random variables supported on the real line, as well as a normally distributed random variable $X$ (whose mean and variance are known but irrelevant). I know that the moments of the $X_n$ converge to the corresponding moment of $X$, that is, for every $k\ge1$, $$\lim_{n\to\infty} \mu_k'(X_n) = \mu_k(X).$$ I need to conclude that the $X_n$ converge to $X$ in distribution.

I believe that this is a standard fact in probability, and I would like an excellent source (including a clear statement and proof) for this fact, to cite in a paper I'm writing. (The application is to number theory, which is why I added the probabilistic-number-theory tag.) I also believe that this conclusion holds for many, but not all, random variables $X$ and not just a normally distributed one; I'd be happy for a general statement or one that applies only to a normal variable.

Nominations for a good citing source, anyone?

It's quite easy to prove it, once you estabilish Prokhorov's theorem; namely use boundedness of some moments to conclude that your sequence of distributions is tight and then it suffices to convey everyone that every convergent subsequence of $(X_n)$ converges to $X$ (because convergence in distribution is metrizable), which is easy, because the limit is characterized by its moments. Before that, one needs a lemma stating that convergence in distribution combined with convergence of moments implies that moments converge to the moments of the limit.
• @Ian: it is an assumption of this theorem that distribution of limit is characterized by its moments. Without this, it is clearly false, because we can take two random variables $X$ and $Y$ which have different distributions and equal moments, and then just take $X_{n} \equiv Y$. Jul 26, 2012 at 14:57
This is a famous problem known as the Hamburger moment problem. It is possible though to get the same result for the normal distribution with a much smaller number of assumptions than requiring convergence for all moments with $k\geq1$.