Hausdorff moment problem
In mathematics, the Hausdorff moment problem, named after Felix Hausdorff, asks for necessary and sufficient conditions that a given sequence (m0, m1, m2, ...) be the sequence of moments
of some Borel measure μ supported on the closed unit interval [0, 1]. In the case m0 = 1, this is equivalent to the existence of a random variable X supported on [0, 1], such that E[Xn] = mn.
The essential difference between this and other well-known moment problems is that this is on a
Completely monotonic sequences
In 1921, Hausdorff showed that (m0, m1, m2, ...) is such a moment sequence if and only if the sequence is completely monotonic, that is, its difference sequences satisfy the equation
for all n, k ≥ 0. Here, Δ is the
The necessity of this condition is easily seen by the identity
which is non-negative since it is the
See also
- Absolutely and completely monotonic functions and sequences
- Total monotonicity
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.
- Feller, W. "An Introduction to Probability Theory and Its Applications", volume II, John Wiley & Sons, 1971.
- Shohat, J.A.; Tamarkin, J. D. The Problem of Moments, American mathematical society, New York, 1943.