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

More generally, one may consider

.

Introduction

is a measure on the real line. 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 is allowed to be the whole real line; the Stieltjes moment problem; and the Hausdorff moment problem for a bounded interval, which without loss of generality.

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 is the complex unit circle instead of the real line.

Existence

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

should be positive semi-definite.

that sends a polynomial

to

, then evidently

Vice versa, if () holds, one can apply the M. Riesz extension theorem to a functional on the space of continuous functions with compact support), so that

By the Riesz representation theorem, (, such that

.

is equivalent to (, one can reformulate () as a condition on Hankel matrices.

Uniqueness (or determinacy)

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. For the problem on an infinite interval, uniqueness is a more delicate question. There are distributions, such as log-normal distributions, which have finite moments for all the positive integers but where other distributions have the same moments.

Formal solution

When the solution exists, it can be formally written using derivatives of the Dirac delta function as

1. .

The expression can be derived by taking the inverse Fourier transform of its characteristic function.

Variations

An important variation is the truncated moment problem, which studies the properties of measures with fixed first moments (for a finite ). Results on the truncated moment problem have numerous applications to extremal problems, optimisation and limit theorems in probability theory.

Probability

The moment problem has applications to probability theory. The following is commonly used:

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:

See also

• Carleman's condition

• Hamburger moment problem

• Hankel matrix

• Hausdorff moment problem

• Moment (mathematics)

• Stieltjes moment problem

• Trigonometric moment problem

Notes

References

• (translated from the Russian by N. Kemmer)