Stieltjes moment problem

In mathematics, the Stieltjes moment problem, named after Thomas Joannes Stieltjes, seeks necessary and sufficient conditions that a sequence { &mu;n, : n = 0, 1, 2, ... } be of the form


 * $$\mu_n=\int_0^\infty x^n\,dF(x)\,$$

for some nondecreasing function F.

The essential difference between this and other well-known moment problems is that this is on a half-line [ 0, &infin; ), whereas in the Hausdorff moment problem one considers a bounded interval [0, 1], and in the Hamburger moment problem one considers the whole line (&minus;&infin;, &infin;).

Let


 * $$\Delta_n=\left[\begin{matrix}

1 & \mu_1 & \mu_2 & \cdots & \mu_{n}   \\ \mu_1 & \mu_2 & \mu_3 & \cdots & \mu_{n+1} \\ \mu_2& \mu_3 & \mu_4 & \cdots & \mu_{n+2} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \mu_{n} & \mu_{n+1} & \mu_{n+2} & \cdots & \mu_{2n} \end{matrix}\right].$$

and


 * $$\Delta_n^{(1)}=\left[\begin{matrix}

\mu_1 & \mu_2 & \mu_3 & \cdots & \mu_{n+1}   \\ \mu_2 & \mu_3 & \mu_4 & \cdots & \mu_{n+2} \\ \mu_3 & \mu_4 & \mu_5 & \cdots & \mu_{n+3} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \mu_{n+1} & \mu_{n+2} & \mu_{n+3} & \cdots & \mu_{2n+1} \end{matrix}\right].$$

Then { &mu;n : n = 1, 2, 3, ... } is a moment sequence of some probability distribution on $$[0,\infty)$$ with infinite support if and only if for all n, both


 * $$\det(\Delta_n) > 0\ \mathrm{and}\ \det\left(\Delta_n^{(1)}\right) > 0.$$

{ &mu;n : n = 1, 2, 3, ... } is a moment sequence of some probability distribution on $$[0,\infty)$$ with finite support of size m if and only if for all $$n \leq m$$, both


 * $$\det(\Delta_n) > 0\ \mathrm{and}\ \det\left(\Delta_n^{(1)}\right) > 0.$$

and for all larger $$n$$


 * $$\det(\Delta_n) = 0\ \mathrm{and}\ \det\left(\Delta_n^{(1)}\right) = 0.$$

The solution is unique if there are constants C and D such that for all n, |&mu;n|&le; CDn(2n)! .