Example of sequence that is not a moment sequence

Question

Can anyone please give me tips on how to show that the sequence $s =(1,1,1,1,0,0,\cdots)$ is not a moment sequence even though this sequence $s$ is positive definite. I know by Hamburger's theorem that a sequence is a moment sequence if and only if it is positive definite. In other word, positive definite sequences and moment sequences are the same but the question says show that it is not a moment sequence.

Answer 1

By the Cauchy-Schwarz inequality, a moment sequence has to be midpoint-log-convex: $$ \mathbb{E}[|X|^n]\cdot\mathbb{E}[|X|^{n+2}]\geq \mathbb{E}[|X|^{n+1}]^2 $$ i.e. has to fulfill a reversed Newton's inequality.
The sequence $1,1,\color{green}{1},\color{red}{1},\color{green}{0},0,0,\ldots$ has an evident issue.

You may also notice that the matrix $$ M = \begin{pmatrix} 1& 1 & 1 & 1 \\ 1 & \color{purple}{1} & \color{purple}{1} & 0 \\ 1 & \color{purple}{1} & \color{purple}{0} &0 \\ 1 & 0 & 0 & 0 \end{pmatrix}$$ is not positive definite since $-1$ is one of its eigenvalues / since the purple minor has determinant $-1$.