We are going to solve the truncated moment problem in this post. The theorem we are going to establish is more general than the original problem itself. The following theorem is a bit abstract, you can skip to Corollary 2 to see what the truncated moment problem is and why it has a generalization in the form of Theorem 1.
Theorem 1 Suppose is a random transformation from a probability space to a measurable space where each singleton set of is in . Let each be a real valued (Borel measurable) function with its domain to be , . Given
and they are all finite, there exists a random variable such that takes no more than values in , and
(If you are not familiar with terms Borel measurable, measurable space and sigma-algebras , then just ignore these. I put these term here just to make sure the that the theorem is rigorous enough.)
Let me parse the theorem for you. Essentially, the theorem is trying to say that given many expectations, no matter what kind of source the randomness comes from, i.e., what is, we can always find a finite valued random variable (which is in the theorem) that achieves the same expectation.
To have a concrete sense of what is going on, consider the following Corollary of Theorem 1. It is the original truncated moment problem.
there exists a real valued discrete random variable which takes no more than values in and its first moments are the same as , i.e.,
This original truncated moment problem is asking that given the (uncentered) moments, can we always find a finite discrete random variable that matches all the moments. It should be clear that is a simple consequence of Theorem 1 by letting and .
There is also a multivariate version of truncated moment problem which can also be regarded as a special case of Theorem 1.
Corollary 3 (Truncated Moment Problem, Multivariate Version) For any real random vector and its all th order moments are finite, i.e.,
for any . Each here is a nonnegative integer. The total number of moments in this case is . Then there is a real random vector such that it takes no more than values, and
Though the form of Theorem 1 is quite general and looks scary, it is actually a simple consequence of the following lemma and the use of convex hull.
The above proposition is trivially true if is closed or takes only finitely many value. But it is true that is only assumed to be convex. We will show it in this post.
We are now ready to show Theorem 1.
Proof of Theorem 1: Consider the set
The convex hull of this set is
Now take the random variable which takes value only in , by Lemma 4 of convex set, we know that
Note that every element in has a FINITE representation in terms of s!
This means we can find , and such that
Since each for some , we can simply take the distribution of to be
Finally, apply the theorem of Caratheodory to conclude that .