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Moment problem

Inmathematics, a moment problem arises as the result of trying to invert the mapping that takes a measure ${\displaystyle \mu }$ to the sequence of moments

${\displaystyle m_{n}=\int _{-\infty }^{\infty }x^{n}\,d\mu ($x)\,.}

More generally, one may consider

${\displaystyle m_{n}=\int _{-\infty }^{\infty }M_{n}($x)\,d\mu (x)\,.}

for an arbitrary sequence of functions ${\displaystyle M_{n}}$.

Introduction[edit]

In the classical setting, ${\displaystyle \mu }$ is a measure on the real line, and ${\displaystyle M}$ is the sequence ${\displaystyle \{x^{n}:n=1,2,\dotsc \}}$. 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 supportof${\displaystyle \mu }$ is allowed to be the whole real line; the Stieltjes moment problem, for ${\displaystyle [0,\infty )}$; and the Hausdorff moment problem for a bounded interval, which without loss of generality may be taken as ${\displaystyle [0,1]}$.

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 μ is the complex unit circle instead of the real line.^[1]

Existence[edit]

A sequence of numbers ${\displaystyle m_{n}}$ is the sequence of moments of a measure ${\displaystyle \mu }$ if and only if a certain positivity condition is fulfilled; namely, the Hankel matrices ${\displaystyle H_{n}}$,

${\displaystyle (H_{n})_{ij}=m_{i+j}\,,}$

should be positive semi-definite. This is because a positive-semidefinite Hankel matrix corresponds to a linear functional ${\displaystyle \Lambda }$ such that ${\displaystyle \Lambda (x^{n})=m_{n}}$ and ${\displaystyle \Lambda (f^{2})\geq 0}$ (non-negative for sum of squares of polynomials). Assume ${\displaystyle \Lambda }$ can be extended to ${\displaystyle \mathbb {R} [$x]^{*}} . In the univariate case, a non-negative polynomial can always be written as a sum of squares. So the linear functional ${\displaystyle \Lambda }$ 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 ${\displaystyle \Lambda (x^{n})=\int _{-\infty }^{\infty }x^{n}d\mu }$. A condition of similar form is necessary and sufficient for the existence of a measure ${\displaystyle \mu }$ supported on a given interval ${\displaystyle [a,b]}$.

One way to prove these results is to consider the linear functional ${\displaystyle \varphi }$ that sends a polynomial

${\displaystyle P($x)=\sum _{k}a_{k}x^{k}}

to

${\displaystyle \sum _{k}a_{k}m_{k}.}$

If${\displaystyle m_{k}}$ are the moments of some measure ${\displaystyle \mu }$ supported on ${\displaystyle [a,b]}$, then evidently

${\displaystyle \varphi ($P)\geq 0} for any polynomial ${\displaystyle P}$ that is non-negative on ${\displaystyle [a,b]}$.

(1)

Vice versa, if (1) holds, one can apply the M. Riesz extension theorem and extend ${\displaystyle \varphi }$ to a functional on the space of continuous functions with compact support ${\displaystyle C_{c}([a,b])}$), so that

${\displaystyle \varphi ($f)\geq 0} for any ${\displaystyle f\in C_{c}([a,b]),\;f\geq 0.}$

(2)

By the Riesz representation theorem, (2) holds iff there exists a measure ${\displaystyle \mu }$ supported on ${\displaystyle [a,b]}$, such that

${\displaystyle \varphi ($f)=\int f\,d\mu }

for every ${\displaystyle f\in C_{c}([a,b])}$.

Thus the existence of the measure ${\displaystyle \mu }$ is equivalent to (1). Using a representation theorem for positive polynomials on ${\displaystyle [a,b]}$, one can reformulate (1) as a condition on Hankel matrices.^[2][3]

Uniqueness (or determinacy)[edit]

The uniqueness of ${\displaystyle \mu }$ 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 functionson${\displaystyle [0,1]}$. For the problem on an infinite interval, uniqueness is a more delicate question.^[4] 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[edit]

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

${\displaystyle d\mu ($x)=\rho (x)dx,\quad \rho (x)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}\delta ^{(n)}(x)m_{n}} .

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

Variations[edit]

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]

Probability[edit]

The moment problem has applications to probability theory. The following is commonly used:^[5]

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[edit]

• Carleman's condition
• Hamburger moment problem
• Hankel matrix
• Hausdorff moment problem
• Moment (mathematics)
• Stieltjes moment problem
• Trigonometric moment problem

Notes[edit]

References[edit]

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