Moment problem
In mathematics, a moment problem arises as the result of trying to invert the mapping that takes a measure μ to the sequences of moments
More generally, one may consider
for an arbitrary sequence of functions Mn.
Contents
Introduction
In the classical setting, μ is a measure on the real line, and M is the sequence { xn : n = 0, 1, 2, ... }. In this form the question appears in probability theory, asking whether there is a probability measure having specified mean, variance and so on, and whether it is unique.
There are three named classical moment problems: the Hamburger moment problem in which the support of μ is allowed to be the whole real line; the Stieltjes moment problem, for [0, +∞); and the Hausdorff moment problem for a bounded interval, which without loss of generality may be taken as [0, 1].
Existence
A sequence of numbers mn is the sequence of moments of a measure μ if and only if a certain positivity condition is fulfilled; namely, the Hankel matrices Hn,
should be positive semi-definite.[clarification needed] A condition of similar form is necessary and sufficient for the existence of a measure supported on a given interval [a, b].
One way to prove these results is to consider the linear functional that sends a polynomial
to
If mkn are the moments of some measure μ supported on [a, b], then evidently
-
φ(P) ≥ 0 for any polynomial P that is non-negative on [a, b].
(1)
Vice versa, if (1) holds, one can apply the M. Riesz extension theorem and extend to a functional on the space of continuous functions with compact support C0([a, b]), so that
-
(2)
such that ƒ ≥ 0 on [a, b].
By the Riesz representation theorem, (2) holds iff there exists a measure μ supported on [a, b], such that
for every ƒ ∈ C0([a, b]).
Thus the existence of the measure is equivalent to (1). Using a representation theorem for positive polynomials on [a, b], one can reformulate (1) as a condition on Hankel matrices.
See Shohat & Tamarkin 1943 and Krein & Nudelman 1977 for more details.
Uniqueness (or determinacy)
The uniqueness of μ in the Hausdorff moment problem follows from the Weierstrass approximation theorem, which states that polynomials are dense under the uniform norm in the space of continuous functions on [0, 1]. For the problem on an infinite interval, uniqueness is a more delicate question; see Carleman's condition, Krein's condition and Akhiezer (1965).
Variations
An important variation is the truncated moment problem, which studies the properties of measures with fixed first k moments (for a finite k). Results on the truncated moment problem have numerous applications to extremal problems, optimisation and limit theorems in probability theory. See also: Chebyshev–Markov–Stieltjes inequalities and Krein & Nudelman 1977.
See also
- Stieltjes moment problem
- Hamburger moment problem
- Hausdorff moment problem
- Moment (mathematics)
- Carleman's condition
- Hankel matrix
References
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