moment problem

In the classical setting, $\backslash mu$ is a measure on the real line, and $M$ is the sequence $\backslash \{x^n\; :\; n=1,2,\backslash dotsc\backslash \}$. 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 $\backslash mu$ is allowed to be the whole real line; the Stieltjes moment problem, for $[0,\backslash infty)$; 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 {{math|μ}} is the complex unit circle instead of the real line.{{sfn | Schmüdgen | 2017 | p=257}}

A sequence of numbers $m\_n$ is the sequence of moments of a measure $\backslash mu$ if and only if a certain positivity condition is fulfilled; namely, the Hankel matrices $H\_n$,

:$(H\_n)\_\{ij\}\; =\; m\_\{i+j\}\backslash ,,$

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

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

:$P(x)\; =\; \backslash sum\_k\; a\_k\; x^k$

to

:$\backslash sum\_k\; a\_k\; m\_k.$

If $m\_k$ are the moments of some measure $\backslash mu$ supported on $[a,b]$, then evidently

{{NumBlk|::|$\backslash varphi(P)\; \backslash ge\; 0$ for any polynomial $P$ that is non-negative on $[a,b]$.|{{EquationRef|1}}}}

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

{{NumBlk|::|$\backslash varphi(f)\; \backslash ge\; 0$ for any $f\; \backslash in\; C\_c([a,b]),\backslash ;f\backslash ge\; 0.$|{{EquationRef|2}}}}

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

:$\backslash varphi(f)\; =\; \backslash int\; f\; \backslash ,\; d\backslash mu$

for every $f\; \backslash in\; C\_c([a,b])$.

Thus the existence of the measure $\backslash mu$ is equivalent to ({{EquationNote|1}}). Using a representation theorem for positive polynomials on $[a,b]$, one can reformulate ({{EquationNote|1}}) as a condition on Hankel matrices.{{sfn|Shohat|Tamarkin|1943}}{{sfn|Kreĭn |Nudel′man|1977}}

{{See also| Carleman's condition|Krein's condition}}

The uniqueness of $\backslash mu$ 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.{{sfn|Akhiezer|1965}} 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.

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

:$d\backslash mu(x)\; =\; \backslash rho(x)\; dx,\; \backslash quad\; \backslash rho(x)\; =\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{(-1)^n\}\{n!\}\backslash delta^\{(n)\}(x)m\_n$

.

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

{{See also|Chebyshev–Markov–Stieltjes inequalities}}

An important variation is the truncated moment problem, which studies the properties of measures with fixed first {{math| k}} moments (for a finite {{math| k}}). Results on the truncated moment problem have numerous applications to extremal problems, optimisation and limit theorems in probability theory.{{sfn|Kreĭn |Nudel′man|1977}}

The moment problem has applications to probability theory. The following is commonly used:{{Cite web |last=Sodin |first=Sasha |date=March 5, 2019 |title=The classical moment problem |url=https://webspace.maths.qmul.ac.uk/a.sodin/teaching/moment/clmp.pdf |url-status=live |archive-url=https://web.archive.org/web/20220701072907/https://webspace.maths.qmul.ac.uk/a.sodin/teaching/moment/clmp.pdf |archive-date=1 Jul 2022}}

{{Math theorem|name=Theorem (Fréchet-Shohat)|note=|math_statement=

If $\backslash mu$ is a determinate measure (i.e. its moments determine it uniquely), and the measures $\backslash mu\_n$ are such that $$$$

\forall k \geq 0 \quad \lim _{n \rightarrow \infty} m_k\left[\mu_n\right]=m_k[\mu],

then $\backslash mu\_n\; \backslash rightarrow\; \backslash mu$ in distribution.

}}

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:

{{Math theorem

| name = Corollary

| note =

| math_statement = If a sequence of probability distributions $\backslash nu\_n$ satisfy $$m\_\{2k\}[\backslash nu\_n]\; \backslash to\; \backslash frac\{(2k)!\}\{2^k\; k!\};\; \backslash quad\; m\_\{2k+1\}[\backslash nu\_n]\; \backslash to\; 0$$ then $\backslash nu\_n$ converges to $N(0,\; 1)$ in distribution.

}}

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

{{Reflist}}

• {{cite book | last1 = Shohat | first1 = James Alexander | first2 = Jacob D. | last2 = Tamarkin | authorlink2 = Jacob Tamarkin | title = The Problem of Moments | publisher = American mathematical society | location = New York | year = 1943 |isbn=978-1-4704-1228-9}}
• {{cite book | last1 = Akhiezer | first1 = Naum I. | authorlink1 = Naum Akhiezer | title = The classical moment problem and some related questions in analysis | url = https://archive.org/details/classicalmomentp0000akhi | url-access = registration | publisher = Hafner Publishing Co. |location = New York | year = 1965 }} (translated from the Russian by N. Kemmer)
• {{cite book | last1=Kreĭn | first1=M. G. | last2=Nudel′man | first2=A. A. | series=Translations of Mathematical Monographs | title=The Markov Moment Problem and Extremal Problems | publisher=American Mathematical Society | publication-place=Providence, Rhode Island | year=1977 | isbn=978-0-8218-4500-4 | issn=0065-9282 | doi=10.1090/mmono/050}}
• {{cite book | last=Schmüdgen | first=Konrad | series= Graduate Texts in Mathematics|title=The Moment Problem | publisher=Springer International Publishing | publication-place=Cham | year=2017 | volume=277 | isbn=978-3-319-64545-2 | issn=0072-5285 | doi=10.1007/978-3-319-64546-9}}

Category:Mathematical analysis

Category:Hilbert spaces

Category:Probability problems

Category:Moments (mathematics)

Category:Mathematical problems

Category:Real algebraic geometry

Category:Optimization in vector spaces