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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?

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  • $\begingroup$ Your question is too unspecific.In which field of research do you need the answers? $\endgroup$ – Dieter Kadelka Feb 17 at 20:42
  • $\begingroup$ What is the region for $K=2,3,4$? $\endgroup$ – Seva Feb 18 at 9:37
  • $\begingroup$ I'm a bit confused -- do you want to test if it is possible for there to exist any $X$ with these moments, or are you also given a particular $X$ (say you are given i.i.d. observations) and you want to test if it has these moments? $\endgroup$ – usul Feb 18 at 12:19
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    $\begingroup$ You are aware of the k infinite versions of this problem ? EG, en.wikipedia.org/wiki/Stieltjes_moment_problem $\endgroup$ – mike Feb 18 at 16:21
  • $\begingroup$ @usul I am interested in the former, e.g., does there exist any positive real $X$ with these moments. I have updated the question to clarify it slightly. $\endgroup$ – hwlin Feb 18 at 17:10
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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.

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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.

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