Hausdorff moment problem

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In mathematics, the Hausdorff moment problem, named after Felix Hausdorff, asks for necessary and sufficient conditions that a given sequence { μn : 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, supported on the closed unit interval [0, 1].

In 1921, Hausdorff showed that { μn : 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.

[edit] References

  • Hausdorff, F. "Summationsmethoden und Momentfolgen. I." Mathematische Zeitschrift 9, 74-109, 1921.
  • Hausdorff, F. "Summationsmethoden und Momentfolgen. II." Mathematische Zeitschrift 9, 280-299, 1921.
  • Shohat, James Alexander, The Problem of Moments, American mathematical society, New York, 1943.

[edit] External links

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