Moment problem explained

\mu to the sequence of momentsm _{n} = \int ^{inf ty} _{-inf ty} x ^{n} d\mu(x).

More generally, one may consider

m _{n} = \int ^{inf ty} _{-inf ty} M _{n(x)} d\mu(x). for an arbitrary sequence of functions M _{n} .

Introduction

In the classical setting,

\mu is a measure on the real line, and M is the sequence \{x ^{n} : n =1 , 2 , ... c\} . 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

\mu is allowed to be the whole real line; the Stieltjes moment problem, for [ 0 , inf ty) ; and the Hausdorff moment problem for a bounded interval, which without loss of generality may be taken as [ 0 , 1 ] .

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

A sequence of numbers

m _{n} is the sequence of moments of a measure \mu if and only if a certain positivity condition is fulfilled; namely, the Hankel matrices H _{n} ,(H _{n)} _{ij} = m _{i+j} ,

should be positive semi-definite. This is because a positive-semidefinite Hankel matrix corresponds to a linear functional

Λ such that Λ (x ^{n)} = m _{n} and Λ (f ^{2)} \geq 0 (non-negative for sum of squares of polynomials). Assume Λ can be extended to R [x] ^{*} . In the univariate case, a non-negative polynomial can always be written as a sum of squares. So the linear functional Λ 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 Λ (x ^{n)} = \int ^{inf ty} _{-inf ty} x ^{n} d \mu . A condition of similar form is necessary and sufficient for the existence of a measure \mu supported on a given interval [a,b] .

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

\varphi that sends a polynomialP(x) = \sum _{k} a _{k} x ^{k}

to

\sum _{k} a _{k} m _{k.}

If

m _{k} are the moments of some measure \mu supported on [a,b] , then evidently

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

\varphi to a functional on the space of continuous functions with compact support C _{c([a,b])} ), so that

By the Riesz representation theorem, holds iff there exists a measure

\mu supported on [a,b] , such that\varphi(f) = \int f d\mu

for every

f \in C _{c([a,b])} .

Thus the existence of the measure

\mu is equivalent to . Using a representation theorem for positive polynomials on [a,b] , one can reformulate as a condition on Hankel matrices.

Uniqueness (or determinacy)

See also: Carleman's condition and Krein's condition. The uniqueness of

\mu 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. 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

d\mu(x) = \rho(x) dx, \rho(x) = \sum ^{inf ty} _{n =0} (( -1 ) ^{n})/(n!) \delta ^{(n)} (x)m _{n} .The expression can be derived by taking the inverse Fourier transform of its characteristic function.

Variations

See also: Chebyshev–Markov–Stieltjes inequalities. 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:^[1]

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

References

• Book: Shohat . James Alexander . Jacob D. . Tamarkin . Jacob Tamarkin . The Problem of Moments . American mathematical society . New York . 1943 . 978-1-4704-1228-9.
• Book: Akhiezer . Naum I. . Naum Akhiezer . The classical moment problem and some related questions in analysis . registration . Hafner Publishing Co. . New York . 1965 . (translated from the Russian by N. Kemmer)
• Book: Kreĭn . M. G. . Nudel′man . A. A. . Translations of Mathematical Monographs . The Markov Moment Problem and Extremal Problems . American Mathematical Society . Providence, Rhode Island . 1977 . 978-0-8218-4500-4 . 0065-9282 . 10.1090/mmono/050.
• Book: Schmüdgen, Konrad . Graduate Texts in Mathematics. The Moment Problem . Springer International Publishing . Cham . 2017 . 277 . 978-3-319-64545-2 . 0072-5285 . 10.1007/978-3-319-64546-9.

Notes and References

1. Web site: Sodin . Sasha . March 5, 2019 . The classical moment problem . live . https://web.archive.org/web/20220701072907/https://webspace.maths.qmul.ac.uk/a.sodin/teaching/moment/clmp.pdf . 1 Jul 2022.