# 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?

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–  Qiaochu Yuan Jul 24 '12 at 0:05
@Qiaouchu's link has a typo. Here is the right link: en.wikipedia.org/wiki/Method_of_moments_(probability_theory) –  Igor Rivin Jul 24 '12 at 14:31
@Igor's link has the same typo. Third time's a [charm](en.wikipedia.org/wiki/Method_of_moments_(probability_theory%29)? –  Erick Wong Jul 31 '12 at 4:09

It is theorem 30.2 in Billingsley's Probability and Measure (I own a second Polish edition, so numbering may differ a little).

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.

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The English editions only have 24 chapters, so I'm not sure what result this is. It seems to me that the difficult part is showing that the limit is characterised by its moments (which is false for certain distributions). –  Ian Morris Jul 26 '12 at 10:53
@Ian: The English edition I own (third edition, 1995) has 38 sections grouped into 7 chapters. The theorem Mateusz refers to is also Theorem 30.2 on p. 390 of my copy. –  Mark Meckes Jul 26 '12 at 13:37
@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 $X_{n} \equiv Y$. –  Mateusz Wasilewski Jul 26 '12 at 14:57
Oops, I was looking at the wrong book by Billingsley! The result required to show that the normal distribution is characterised by its moments is also in the book Mateusz suggests, as Theorem 30.1. –  Ian Morris Jul 26 '12 at 15:04

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 $k\geq1$.

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