# 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, 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.