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)