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 [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
be a Hankel matrix, and
Then { mn : n = 1, 2, 3, ... } is a moment sequence of some measure on with infinite support if and only if for all n, both
{ mn : n = 1, 2, 3, ... } 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.. Reprinted in . Reprinted in
References