# When does the limit of moments of multivariate distributions determine the limit distribution?

Hello I'm sorry if this question is trivial but I haven't been able to find an answer. I'm trying to show that a sequence of distributions on $\mathbb{R}^n$ converges to the normal distribution by showing that the moments of the distributions converge to those of the normal distribution. Is this sufficient under appropriate assumptions? Thanks

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Yes, without any additional assumptions — the relevant technical conditions are satisfied because your limit distribution is normal. For sufficiency in the univariate case, see any probability textbook that covers the method of moments, for example section 30 of Billingsley's Probability and Measure. The multivariate case follows from the univariate case by the Cramér–Wold device, see for example section 29 of Billingsley.

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Also, as for whether the question is trivial, it's easy and well-known folklore, but it's surprisingly hard to find written down. – Mark Meckes Dec 13 '12 at 17:35
Thanks a lot! It's nice not to be working towards a dead end:-) – NA007 Dec 13 '12 at 18:51