Talk:Stieltjes moment problem

Positive Semi-Definiteness

Shouldn't the condition

$\det(\Delta _{n})>0$

(and possibly also $\det \left(\Delta _{n}^{(1)}\right)>0$) be relaxed to

$\det(\Delta _{n})\geq 0$

(and possibly $\det \left(\Delta _{n}^{(1)}\right)\geq 0$)? At least

$\det(\Delta _{n})>0$

would imply that the Variance of a random variable is always strictly larger than zero, which for instance is not the case for the degenerate Dirac-distribution. — Preceding unsigned comment added by Imperialfists (talk • contribs) 19:57, 20 February 2016 (UTC)

This is a slight issue with the finite support statement, and you have identified an $n=1$ counterexample if $\mathbb {P} (X=0)=1$ so $\det(\Delta _{1})=1$ but $\det \left(\Delta _{1}^{(1)}\right)=0$. Another counterexample happens with $n=2$ and a Bernoulli random variable if $\mathbb {P} (X=0)=1-p$ and $\mathbb {P} (X=1)=p$ in which case $m_{0}=1,m_{1}=m_{2}=m_{3}=p$ and so $\det(\Delta _{2})=p(1-p)>0$ but $\det \left(\Delta _{2}^{(1)}\right)=0$, and I would expect this kind of counterexample to continue for all n whenever any of the n values in the support is 0 with positive probability. Rumping (talk) 01:55, 28 January 2021 (UTC)