Editing Moment problem

{{Use American English|date = March 2019}}
{{Short description|Trying to map moments to a measure that generates them}}
[[File:Standard_deviation_diagram.svg|thumb|Example: Given the mean and variance <math>\sigma^2</math> (as well as all further  [[cumulant]]s equal 0) the [[normal distribution]] is the distribution solving the moment problem.]]
In [[mathematics]], a '''moment problem''' arises as the result of trying to invert the mapping that takes a [[measure (mathematics)|measure]] <math>\mu</math> to the sequence of [[Moment (mathematics)|moment]]s

:<math>m_n = \int_{-\infty}^\infty x^n \,d\mu(x)\,.</math>

More generally, one may consider
:<math>m_n = \int_{-\infty}^\infty M_n(x) \,d\mu(x)\,.</math>
for an arbitrary sequence of functions <math>M_n</math>.

== Introduction ==

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

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|''&mu;''}} is the [[Circle group|complex unit circle]] instead of the real line.{{sfn | Schmüdgen | 2017 | p=257}}

==Existence==

A sequence of numbers <math>m_n</math> is the sequence of moments of a measure <math>\mu</math> if and only if a certain positivity condition is fulfilled; namely, the [[Hankel matrices]] <math>H_n</math>,

:<math>(H_n)_{ij} = m_{i+j}\,,</math>

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

One way to prove these results is to consider the linear functional <math>\varphi</math> that sends a polynomial

:<math>P(x) = \sum_k a_k x^k </math>

to

:<math>\sum_k a_k m_k.</math>

If <math>m_k</math> are the moments of some measure <math>\mu</math> supported on <math>[a,b]</math>, then evidently

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

Vice versa, if ({{EquationNote|1}}) holds, one can apply the [[M. Riesz extension theorem]] and extend <math>\varphi</math> to a functional on the [[Function space#Functional analysis|space of continuous functions with compact support]]  <math>C_c([a,b])</math>), so that

{{NumBlk|::|<math>\varphi(f) \ge 0</math> for any  <math>f \in C_c([a,b]),\;f\ge 0.</math>|{{EquationRef|2}}}}

By the [[Riesz representation theorem#The representation theorem for linear functionals on Cc.28X.29|Riesz representation theorem]], ({{EquationNote|2}}) holds iff there exists a measure <math>\mu</math> supported on <math>[a,b]</math>, such that

:<math> \varphi(f) = \int f \, d\mu</math>

for every <math>f \in C_c([a,b])</math>.

Thus the existence of the measure <math>\mu</math> is equivalent to ({{EquationNote|1}}). Using a representation theorem for positive polynomials on <math>[a,b]</math>, <!-- This is due to Riesz or Fejer (or maybe both); a ref. is needed (maybe Szego's book?) --> one can reformulate ({{EquationNote|1}}) as a condition on Hankel matrices.{{sfn|Shohat|Tamarkin|1943}}{{sfn|Kreĭn |Nudel′man|1977}}

== Uniqueness (or determinacy) ==
{{See also| Carleman's condition|Krein's condition}}
The uniqueness of <math>\mu</math> in the Hausdorff moment problem follows from the [[Weierstrass approximation theorem]], which states that [[polynomial]]s are [[dense set|dense]] under the [[uniform norm]] in the space of [[continuous functions]] on <math>[0,1]</math>. For the problem on an infinite interval, uniqueness is a more delicate question.{{sfn|Akhiezer|1965}} There are distributions, such as [[log-normal distribution]]s, 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
:<math> d\mu(x) = \rho(x) dx, \quad \rho(x) = \sum_{n=0}^\infty \frac{(-1)^n}{n!}\delta^{(n)}(x)m_n
</math>.
The expression can be derived by taking the inverse Fourier transform of its [[Characteristic function (probability theory)|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 {{math| ''k''}} moments (for a finite {{math| ''k''}}). Results on the truncated moment problem have numerous applications to [[extremal problem]]s, optimisation and limit theorems in [[probability theory]].{{sfn|Kreĭn |Nudel′man|1977}}

== Probability ==

The moment problem has applications to probability theory. The following is commonly used:<ref>{{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}}</ref>

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

If <math display="inline">\mu</math> is a determinate measure (i.e. its moments determine it uniquely), and the measures <math display="inline">\mu_n</math> are such that <math display="block">
      \forall k \geq 0 \quad \lim _{n \rightarrow \infty} m_k\left[\mu_n\right]=m_k[\mu],
      </math> then <math display="inline">\mu_n \rightarrow \mu</math> 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 <math display="inline">\nu_n</math> satisfy <math display="block">m_{2k}[\nu_n] \to \frac{(2k)!}{2^k k!}; \quad m_{2k+1}[\nu_n] \to 0</math> then <math display="inline">\nu_n</math> converges to <math display="inline">N(0, 1)</math> in distribution.
}}

== See also ==
*[[Carleman's condition]]
*[[Hamburger moment problem]]
*[[Hankel matrix]]
*[[Hausdorff moment problem]]
*[[Moment (mathematics)]]
*[[Stieltjes moment problem]]
*[[Trigonometric moment problem]]

==Notes==
{{Reflist}}

==References==
*{{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]]