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

${\displaystyle m_{n}=\int _{0}^{\infty }x^{n}\,d\mu (x)}$

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

${\displaystyle \Delta _{n}=\left[{\begin{matrix}m_{0}&m_{1}&m_{2}&\cdots &m_{n}\\m_{1}&m_{2}&m_{3}&\cdots &m_{n+1}\\m_{2}&m_{3}&m_{4}&\cdots &m_{n+2}\\\vdots &\vdots &\vdots &\ddots &\vdots \\m_{n}&m_{n+1}&m_{n+2}&\cdots &m_{2n}\end{matrix}}\right]}$

be a Hankel matrix, and

${\displaystyle \Delta _{n}^{(1)}=\left[{\begin{matrix}m_{1}&m_{2}&m_{3}&\cdots &m_{n+1}\\m_{2}&m_{3}&m_{4}&\cdots &m_{n+2}\\m_{3}&m_{4}&m_{5}&\cdots &m_{n+3}\\\vdots &\vdots &\vdots &\ddots &\vdots \\m_{n+1}&m_{n+2}&m_{n+3}&\cdots &m_{2n+1}\end{matrix}}\right].}$

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

${\displaystyle \det(\Delta _{n})>0\ \mathrm {and} \ \det \left(\Delta _{n}^{(1)}\right)>0.}$

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

${\displaystyle \det(\Delta _{n})>0\ \mathrm {and} \ \det \left(\Delta _{n}^{(1)}\right)>0}$

and for all larger ${\displaystyle n}$

${\displaystyle \det(\Delta _{n})=0\ \mathrm {and} \ \det \left(\Delta _{n}^{(1)}\right)=0.}$

Uniqueness

There are several sufficient conditions for uniqueness.

Carleman's condition: The solution is unique if

${\displaystyle \sum _{n\geq 1}m_{n}^{-1/(2n)}=\infty ~.}$

Hardy's criterion: If ${\displaystyle \mu }$ is a probability distribution supported on ${\displaystyle [0,\infty )}$, such that ${\displaystyle \mathbb {E} _{X\sim \mu }[e^{c{\sqrt {X}}}]<\infty }$, then all its moments are finite, and ${\displaystyle \mu }$ is the unique distribution with these moments.^[1][2][3]

References

1. Stoyanov, J.; Lin, G. D. (January 2013). "Hardy's Condition in the Moment Problem for Probability Distributions". Theory of Probability & Its Applications. 57 (4): 699–708. doi:10.1137/S0040585X9798631X. ISSN 0040-585X.
2. Hardy, G. H. (1917). "On Stieltjes' "problème des moments"". Messenger of Mathematics. 46: 175–182.. Reprinted in Hardy, G. H. (1979). Collected Papers of G. H. Hardy. Vol. VII. Oxford: Oxford University Press. pp. 75–83.
3. Hardy, G. H. (1918). "On Stieltjes' "problème des moments" (continued)". Messenger of Mathematics. 47: 81–88.. Reprinted in Hardy, G. H. (1979). Collected Papers of G. H. Hardy. Vol. VII. Oxford: Oxford University Press. pp. 84–91.

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