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The problem is: can you recover a distribution of random variable if you know all its moments?

My first guess was to use moment-generating function (MGF). It is known that if two random variables have the same MGF, then they have equal distributions. So if $M(X) = 1 + \sum_{n = 1}^\infty \mathbb{E} X^n \frac{t^n}{n!}$ converges, then it uniquely corresponds to some distribution; we can apply inverse Laplace transform and get the PDF of X, which solves the problem. But there are some issues:

  1. MGF may not exist.

  2. Even if MGF exists, the approach with Laplace transform works only with continuous random variables.

Can you help me with this problem? Thank you!

esokolov
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  • For 1, if the MGF does not exist in an open neighbourhood of $0$ but all the moments are finite, then there may not be a unique distribution. The lognormal leads to a classic example. See https://stats.stackexchange.com/questions/25010/identity-of-moment-generating-functions – Henry Apr 12 '22 at 00:44

1 Answers1

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The characteristic function may be what you need. It is connected closely to MGF but it always exists and there is one-to-one correspondence between characteristic functions and cumulative distributions. Look for example into quoted Wiki article.

xen
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