This is the second part of the story that I started with the previous post.
Wanting to build a quick counterexample, I wondered if I can create a random variable by solving the moment problem with only finite non-zero moments. In the language of the moment problem, my sequence was such that for all greater than some number . That is, only finite elements of the sequence were non-zero.
Luckily, there’s no need to use Hausdorff’s criterion to judge if this is a good sequence or not. Lyapunov’s inequality for random variables states that if , then
Lyapunov’s inequality can be proved simply by using the fact that is a convex function when . Now, if I set , for any integer, I get that
So, the sequence has to at least satisfy the relation
for all . Clearly, a sequence with finite non-zero elements will not work, as have to be non-negative.