Stieltjes moment problem

In mathematics, the Stieltjes moment problem, named after Thomas Joannes Stieltjes, seeks necessary and sufficient conditions for a sequence (m₀, m₁, m₂, ...) to be of the form

$m_n=\underset{0}{\overset{\mathrm{\infty }}{\int }}{x}^{n}d\mu (x)$

for some measure μ. If such a function μ exists, one asks whether it is unique.

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 (−∞, ∞).

Existence

Let

${\mathrm{\Delta }}_n=\left[\begin{array}{ccccc}{m}_0& m_1& m_2& \cdots & m_n\\ m_1& m_2& m_3& \cdots & m_{n+1}\\ m_2& m_3& m_4& \cdots & m_{n+2}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ m_n& m_{n+1}& m_{n+2}& \cdots & m_{2n}\end{array}\right]$

be a Hankel matrix, and

${\mathrm{\Delta }}_n^{(1)}=\left[\begin{array}{ccccc}{m}_1& m_2& m_3& \cdots & m_{n+1}\\ m_2& m_3& m_4& \cdots & m_{n+2}\\ m_3& m_4& m_5& \cdots & m_{n+3}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ m_{n+1}& m_{n+2}& m_{n+3}& \cdots & m_{2n+1}\end{array}\right].$

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

$\det ({\mathrm{\Delta }}_n)>0\mathrm{a}\mathrm{n}\mathrm{d}\det \left({\mathrm{\Delta }}_n^{(1)}\right)>0.$

{ m_n : n = 1, 2, 3, ... } is a moment sequence of some measure on $[0,\mathrm{\infty })$ with finite support of size m if and only if for all $n\le m$, both

$\det ({\mathrm{\Delta }}_n)>0\mathrm{a}\mathrm{n}\mathrm{d}\det \left({\mathrm{\Delta }}_n^{(1)}\right)>0$

and for all larger $n$

$\det ({\mathrm{\Delta }}_n)=0\mathrm{a}\mathrm{n}\mathrm{d}\det \left({\mathrm{\Delta }}_n^{(1)}\right)=0.$

Uniqueness

There are several sufficient conditions for uniqueness.

Carleman's condition: The solution is unique if

$\sum _{n\ge 1}{m}_n^{-1/(2n)}=\mathrm{\infty }.$

Hardy's criterion: If $\mu$ is a probability distribution supported on $[0,\mathrm{\infty })$, such that ${\mathbb{𝔼}}_{X\sim \mu }[e^{c\sqrt{X}}]<\mathrm{\infty }$, then all its moments are finite, and $\mu$ is the unique distribution with these moments.

References

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