Stieltjes moment problem

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In mathematics, the Stieltjes moment problem, named after Thomas Joannes Stieltjes, seeks necessary and sufficient conditions that a sequence { μn, : 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, ∞), whereas in the Hausdorff moment problem one considers a bounded interval [0, 1], and in the Hamburger moment problem one considers the whole line (−∞, ∞).

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 { μn : 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.

{ μn : 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, |μn|≤ CDn(2n)! (Reed & Simon 1975, p. 341).

[edit] References

  • Reed, Michael & Simon, Barry (1975), Fourier Analysis, Self-Adjointness, vol. 2, Methods of modern mathematical physics, Academic Press, p. 341 (exercise 25), ISBN 0-12-585002-6