Hausdorff moment problem

In mathematics, the Hausdorff moment problem, named after Felix Hausdorff, asks for necessary and sufficient conditions that a given sequence { &mu;n : n = 1, 2, 3, ... } be the sequence of moments


 * $$\mu_n = \operatorname{E}(X^n) = \int_0^1 x^n\,dF(x)\,$$

of some probability distribution, with cumulative distribution function F constant outside the closed unit interval [0, 1]. (This is equivalent to requiring that X take values in [0,1] almost surely.)

The essential difference between this and other well-known moment problems is that this is on a bounded interval, whereas in the Stieltjes moment problem one considers a a half-line [0, ∞), and in the Hamburger moment problem one considers the whole line (−∞, ∞).

In 1921, Hausdorff showed that { &mu;n : n = 1, 2, 3, ... } is such a moment sequence if and only if all of the differences


 * $$\Delta^k \mu_n \,$$

are non-negative, where $$\Delta\,$$ is the difference operator given by


 * $$\Delta \mu_n = \mu_n - \mu_{n+1}.\,$$

For example, it is necessary to have


 * $$\Delta^4 \mu_6 = \mu_6 - 4\mu_7 + 6\mu_8 - 4\mu_9 + \mu_{10} \geq 0.\,$$

When one considers that this is the same as


 * $$\operatorname{E}(X^6(1-X)^4),\,$$

or, generally,


 * $$\Delta^k \mu_n=\operatorname{E}(X^n(1-X)^k)\,$$

then the necessity of these conditions becomes obvious.