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In a version of Levy's theorem we know that if we have a sequence of characteristic functions $\phi_n$, such that $\phi_n\rightarrow \phi$ pointwise, then $\phi$ is the characteristic function of a measure $P$ and $P_n\rightarrow^d P$.

In some theorems, instead of assuming that a sequence of r.v. is identically distributed, we just usually assume that they have some of the moments, until a certain order, equal/limited.

So, is there any theorem that tells how close we are from a distribution, dependent on the order of moments we demand to be equal/limited ?

An old man in the sea.
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  • What do you mean by "how close we are from a distribution," exactly? – Math1000 Nov 19 '15 at 14:32
  • @Math1000 Something like this https://en.wikipedia.org/wiki/Berry%E2%80%93Esseen_theorem , where we can check how far the approximation is from the normal distribution. – An old man in the sea. Nov 19 '15 at 17:43
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    Note that there are distributions which do have the same moments (of all orders), but the distributions do not coincide, see this question: http://math.stackexchange.com/q/1166637/ – saz Nov 19 '15 at 18:27

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