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.

Introduction

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

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. This is because a positive-semidefinite Hankel matrix corresponds to a linear functional such that and . Assume can be extended to. In the univariate case, a non-negative polynomial can always be written as a sum of squares. So the linear functional is positive for all the non-negative polynomials in the univariate case. By Haviland's theorem, the linear functional has a measure form, that is. A condition of similar form is necessary and sufficient for the existence of a measure supported on a given interval .
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 , then evidently
Vice versa, if holds, one can apply the M. Riesz extension theorem and extend to a functional on the space of continuous functions with compact support C0, so that
By the Riesz representation theorem, holds iff there exists a measure μ supported on , such that
for every ƒ ∈ C0.
Thus the existence of the measure is equivalent to. Using a representation theorem for positive polynomials on , one can reformulate as a condition on Hankel matrices.
See and 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 . For the problem on an infinite interval, uniqueness is a more delicate question; see Carleman's condition, Krein's condition and.

Variations

An important variation is the truncated moment problem, which studies the properties of measures with fixed first k moments. 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.