Moments of a positive random variable 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?
 A: This is known as the truncated Stieltjes moment problem, and there is a necessary and sufficient condition taking the form of a semidefinite program. See Section 5 of the classic paper by Curto and Fialkow.
A: Seva asked for the region when $K=2,3,4$
Empirically it seems
$$m_1 > 0$$
$$m_2 > m_1^2$$
$$m_3 > \dfrac{m_2^2}{m_1}$$
$$m_4 > \dfrac{m_3^2+m_2^3-2m_1 m_2 m_3}{m_2-m_1^2}$$
and while it is possible to turn one of these inequalities into an equality for a particular moment, that then fixes every higher moment, with  


*

*$m_1=0 \implies m_2=0$ and $m_3=0$ and $m_4=0$

*$m_2=m_1^2  \implies m_3=m_1^3$ and $m_4=m_1^4$

*$m_3=\frac{m_2^2}{m_1}  \implies m_4=\frac{m_2^3}{m_1^2}$
It also seems empirically that it is possible for find an example for $X$ with the first $K$ given moments where $X$ can take $\lceil (K+1)/2\rceil$ possible non-negative values with associated probabilities (if $K$ is even then one of the values can be $0$), where this example can then give the boundary for the next higher moment. Finding the example involves solving a set of polynomial simultaneous equations.  
