Stieltjes moment problem

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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

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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. [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.