- Hausdorff moment problem
-
In mathematics, the Hausdorff moment problem, named after Felix Hausdorff, asks for necessary and sufficient conditions that a given sequence { mn : n = 0, 1, 2, ... } 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 bounded interval, whereas in the Stieltjes moment problem one considers a half-line [0, ∞), and in the Hamburger moment problem one considers the whole line (−∞, ∞).
In 1921, Hausdorff showed that { mn : n = 0, 1, 2, ... } is such a moment sequence if and only if the sequence is completely monotonic, i.e., its difference sequences satisfy the equation
for all n,k ≤ 0. Here, Δ is the difference operator given by
- (Δm)n = mn + 1 − mn.
The necessity of this condition is easily seen by the identity
which is ≥ 0, being the integral of an almost sure non-negative function. For example, it is necessary to have
See also
- 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", volulme II, John Wiley & Sons, 1971.
- Shohat, J.A.; Tamarkin, J. D. The Problem of Moments, American mathematical society, New York, 1943.
External links
Categories:- Probability theory
- Theory of probability distributions
- Mathematical problems
Wikimedia Foundation. 2010.