Moment problem

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 m_n is the sequence of moments of a measure μ if and only if a certain positivity condition is fulfilled; namely, the Hankel matrices H_n,

should be positive semi-definite. This is because a positive-semidefinite Hankel matrix corresponds to a linear functional Λ such that Λ ( x n ) = m n and Λ ( f 2 ) ≥ 0 (non-negative for sum of squares of polynmials). Assume Λ can be extended to R [ x ] ∗ . In the univariate case, a non-negative polynomial can always be written as a sum of squares. So the linear functional Λ is positive for all the non-negative polynomials in the univariate case. By Haviland theorem, the linear functional has a measure form, that is Λ ( x n ) = ∫ − ∞ ∞ x n d μ . A condition of similar form is necessary and sufficient for the existence of a measure μ supported on a given interval [a, b].

One way to prove these results is to consider the linear functional φ that sends a polynomial

to

If m_kn are the moments of some measure μ supported on [a, b], then evidently

Vice versa, if (1) holds, one can apply the M. Riesz extension theorem and extend ϕ to a functional on the space of continuous functions with compact support C₀([a, b]), so that

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 μ 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.