But moment picking is tricky too.

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 X by solving the moment problem with only finite non-zero moments. In the language of the moment problem, my sequence was such that m_n=0 for all n greater than some number N. 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 X states that if 0<t<s, then

E[|X|^s]^{1/s}\leq E[|X|^t]^{1/t}.

Lyapunov’s inequality can be proved simply by using the fact that x^a is a convex function when a>1. Now, if I set t=n,s=n+1, for any n integer, I get that

E[|X|^n]^{1/n}\leq E[|X|^{n+1}]^{1/(n+1)}.

So, the sequence m_n has to at least satisfy the relation

m_n^{1/n}\leq m_{n+1}^{1/(n+1)}

for all n. Clearly, a sequence with finite non-zero elements will not work, as m_{2k} have to be non-negative.