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

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 <nowiki>[</nowiki>0,&nbsp;&infin;<nowiki>)</nowiki>, whereas in the Hausdorff moment problem one considers a bounded interval [0,&nbsp;1], and in the Hamburger moment problem one considers the whole line (−&infin;,&nbsp;&infin;).

Existence

Let

be a Hankel matrix, and

Then {&nbsp;m_n&nbsp;:&nbsp;n&nbsp;=&nbsp;1,&nbsp;2,&nbsp;3,&nbsp;...&nbsp;} is a moment sequence of some measure on with infinite support if and only if for all n, both

{&nbsp;m_n&nbsp;:&nbsp;n&nbsp;=&nbsp;1,&nbsp;2,&nbsp;3,&nbsp;...&nbsp;} is a moment sequence of some measure on with finite support of size m if and only if for all , both

and for all larger

Uniqueness

There are several sufficient conditions for uniqueness.

Carleman's condition: The solution is unique if

Hardy's criterion: If is a probability distribution supported on , such that , then all its moments are finite, and is the unique distribution with these moments.

References