Hamburger moment problem

In mathematics, the Hamburger moment problem, named after Hans Ludwig Hamburger, is formulated as follows: given a sequence (m₀, m₁, m₂, ...), does there exist a positive Borel measure μ (for instance, the measure determined by the cumulative distribution function of a random variable) on the real line such that

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

In other words, an affirmative answer to the problem means that (m₀, m₁, m₂, ...) is the sequence of moments of some positive Borel measure μ.

The Stieltjes moment problem, Vorobyev moment problem, and the Hausdorff moment problem are similar but replace the real line by ${\displaystyle [0,+\infty )}$ (Stieltjes and Vorobyev; but Vorobyev formulates the problem in the terms of matrix theory), or a bounded interval (Hausdorff).