Moment problem

In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a moment problem arises as the result of trying to invert the mapping that takes a measure

Measure (mathematics)

In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...

 μ to the sequences of moment

Moment (mathematics)

In mathematics, a moment is, loosely speaking, a quantitative measure of the shape of a set of points. The "second moment", for example, is widely used and measures the "width" of a set of points in one dimension or in higher dimensions measures the shape of a cloud of points as it could be fit by...

s


More generally, one may consider
for an arbitrary sequence of functions M_n.

Introduction

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

Real line

In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...

, and M is in the sequence { xⁿ : n = 0, 1, 2, ... } In this form the question appears in probability theory

Probability theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

, asking whether there is a probability measure

Probability measure

In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity...

 having specified mean

Mean

In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....

, variance

Variance

In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...

 and so on, and whether it is unique.

There are three named classical moment problems: the Hamburger moment problem in which the support

Support (mathematics)

In mathematics, the support of a function is the set of points where the function is not zero, or the closure of that set . This concept is used very widely in mathematical analysis...

 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

Without loss of generality

Without loss of generality is a frequently used expression in mathematics...

 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

Positive-definite matrix

In linear algebra, a positive-definite matrix is a matrix that in many ways is analogous to a positive real number. The notion is closely related to a positive-definite symmetric bilinear form ....

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

Vice versa, if holds, one can apply the M. Riesz extension theorem

M. Riesz extension theorem

The M. Riesz extension theorem is a theorem in mathematics, proved by Marcel Riesz during his study of the problem of moments.-Formulation:Let E be a real vector space, F ⊂ E a vector subspace, and let K ⊂ E be a convex cone.A linear functional ...

 and extend to a functional on the space of continuous functions with compact support C₀([a, b]), so that

such that ƒ ≥ 0 on [a, b].

By the Riesz representation theorem, 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 . Using a representation theorem for positive polynomials on [a, b], one can reformulate 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 polynomial

Polynomial

In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

s are dense

Dense set

In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...

 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

Carleman's condition

In mathematics, Carleman's condition is a sufficient condition for the determinacy of the moment problem.-Hamburger moment problem:For the Hamburger moment problem, the theorem, proved by Torsten Carleman, states the following:...

, 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

Probability theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

. See also: Chebyshev–Markov–Stieltjes inequalities and Ref. 3.