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

where M_n(x) is the nth in a list of monomials, for n = 0, 1, 2, 3, ... .

Introduction

In the classical setting, μ is a measure on the real line, and M is in the sequence { xⁿ : n = 0, 1, 2, ... }, giving moments m_n for n = 0, 1, 2, 3, ... . It is in this form that the question would appear in probability theory, of asking to whether there is a probability measure having specified mean, variance and so on.

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

It was realized that this is closely connected to Hilbert spaces and spectral theory. In more concrete terms, there is a condition on a positive measure μ, namely that

for every complex-valued polynomial P(x), unless P vanishes on the support of μ. This gives rise to matrix conditions, necessary on any sequence of moments, namely that certain Hankel matrices are positive semi-definite.

Uniqueness (or determinacy)

The uniqueness of μ in the Hausdorff moment problem follows because polynomials are dense in the uniform norm on [0, 1]. For the problem on an infinite interval, uniqueness is a more delicate question; see Carleman's condition, Krein's condition and Ref. 2.

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 Ref. 3.

See also

• M. Riesz extension theorem

References

1. Shohat, James Alexander; Tamarkin, J. D; The Problem of Moments, American mathematical society, New York, 1943.

2. Akhiezer, N. I., The classical moment problem and some related questions in analysis, translated from the Russian by N. Kemmer, Hafner Publishing Co., New York 1965 x+253 pp.

3. Krein, M. G.; Nudelman, A. A.; The Markov moment problem and extremal problems. Ideas and problems of P. L. Chebyshev and A. A. Markov and their further development. Translated from the Russian by D. Louvish. Translations of Mathematical Monographs, Vol. 50. American Mathematical Society, Providence, R.I., 1977. v+417 pp.

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