# 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

$m_n = \int_{-\infty}^\infty x^n \,d\mu(x)\,.\,\!$

More generally, one may consider

$m_n = \int_{-\infty}^\infty M_n(x) \,d\mu(x)\,.\,\!$

for an arbitrary sequence of functions Mn.

## Introduction

In the classical setting, μ is a measure on the real line, and M is in 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,

$(H_n)_{ij} = m_{i+j}\,,\,\!$

should be positive semi-definite. A condition of similar form is necessary and sufficient for the existence of a measure $\mu$ supported on a given interval [ab].

One way to prove these results is to consider the linear functional $\scriptstyle\varphi$ that sends a polynomial

$P(x) = \sum_k a_k x^k \,\!$

to

$\sum_k a_k m_k.\,\!$

If mkn are the moments of some measure μ supported on [ab], then evidently

φ(P) ≥ 0 for any polynomial P that is non-negative on [ab].

(1)

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

$\qquad \varphi(f) \ge 0\text{ for any } f \in C_0([a,b])$

(2)

such that ƒ ≥ 0 on [ab].

By the Riesz representation theorem, (2) holds iff there exists a measure μ supported on [ab], such that

$\phi(f) = \int f \, d\mu\,\!$

for every ƒ ∈ C0([ab]).

Thus the existence of the measure $\mu$ is equivalent to (1). Using a representation theorem for positive polynomials on [ab], one can reformulate (1) as a condition on Hankel matrices.

See Refs. 1–3. 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 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.