Moment problem

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

to the sequence of moments More generally, one may consider for an arbitrary sequence of functions

is a measure on the real line, and

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

is allowed to be the whole real line; the Stieltjes moment problem, for

; and the Hausdorff moment problem for a bounded interval, which without loss of generality may be taken as

The moment problem also extends to complex analysis as the trigonometric moment problem in which the Hankel matrices are replaced by Toeplitz matrices and the support of μ is the complex unit circle instead of the real line.

is the sequence of moments of a measure

if and only if a certain positivity condition is fulfilled; namely, the Hankel matrices

This is because a positive-semidefinite Hankel matrix corresponds to a linear functional

(non-negative for sum of squares of polynomials).

In the univariate case, a non-negative polynomial can always be written as a sum of squares.

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

One way to prove these results is to consider the linear functional

, then evidently Vice versa, if (1) holds, one can apply the M. Riesz extension theorem and extend

), so that By the Riesz representation theorem, (2) holds iff there exists a measure

Using a representation theorem for positive polynomials on

, one can reformulate (1) as a condition on Hankel matrices.

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.

When the solution exists, it can be formally written using derivatives of the Dirac delta function as The expression can be derived by taking the inverse Fourier transform of its characteristic function.

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.

[3] The moment problem has applications to probability theory.

The following is commonly used:[5] Theorem (Fréchet-Shohat) — If

is a determinate measure (i.e. its moments determine it uniquely), and the measures

By checking Carleman's condition, we know that the standard normal distribution is a determinate measure, thus we have the following form of the central limit theorem: Corollary — If a sequence of probability distributions