Suppose one is handed a list of $K$ numbers, with a claim that these numbers are the first $K$ moments of a *positive* random variable $X$ (meaning there is 0 probability that $X<0$).

What is the strongest possible test that one could run on this list to test this claim? (We do not know any additional information about $X$.) The most obvious thing to check first is that all the moments are positive. A better test would involve checking that Jensen’s inequalities are satisfied. What is the most powerful test?

In general, there is a convex "allowed region” in the $K$-dimensional space of possible moments of $X$. Is there a good way to characterize this space?