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In
mathematics, a moment problem arises as the result of trying to invert the mapping that takes a
measure ''μ'' to the sequences of
moments
:
More generally, one may consider
:
for an arbitrary sequence of functions ''M''
''n''.
Introduction
In the classical setting, μ is a measure on the
real line, and ''M'' is the sequence . In this form the question appears in
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
, asking whether there is a
probability measure having specified
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the '' ari ...
,
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
and so on, and whether it is unique.
There are three named classical moment problems: the
Hamburger moment problem In mathematics, the Hamburger moment problem, named after Hans Ludwig Hamburger, is formulated as follows: given a sequence (''m''0, ''m''1, ''m''2, ...), does there exist a positive Borel measure ''μ'' (for instance, the measure determined by t ...
in which the
support of μ is allowed to be the whole real line; the
Stieltjes moment problem, for
; and the Hausdorff moment problem">, +∞); and the Hausdorff moment problem for a bounded interval, which without loss of generality may be taken as
, 1
Existence
A sequence of numbers ''m''
''n'' is the sequence of moments of a measure ''μ'' if and only if a certain positivity condition is fulfilled; namely, the
Hankel matrices ''H''
''n'',
:
should be
positive semi-definite. This is because a positive-semidefinite Hankel matrix corresponds to a linear functional
such that
and
(non-negative for sum of squares of polynomials). Assume
can be extended to
. In the univariate case, a non-negative polynomial can always be written as a sum of squares. So the linear functional
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
. A condition of similar form is necessary and sufficient for the existence of a measure
supported on a given interval
'a'', ''b''
One way to prove these results is to consider the linear functional
that sends a polynomial
:
to
:
If ''m''
''kn'' are the moments of some measure ''μ'' supported on
'a'', ''b'' then evidently
Vice versa, if () holds, one can apply the
M. Riesz extension theorem
The M. Riesz extension theorem is a theorem in mathematics, proved by Marcel Riesz during his study of the Moment problem, problem of moments.
Formulation
Let E be a real number, real vector space, F\subset E be a vector subspace, and K\subset E ...
and extend
to a functional on the space of continuous functions with compact support ''C''
0(
'a'', ''b'', so that
By the
Riesz representation theorem
:''This article describes a theorem concerning the dual of a Hilbert space. For the theorems relating linear functionals to Measure (mathematics), measures, see Riesz–Markov–Kakutani representation theorem.''
The Riesz representation theorem, ...
, () holds iff there exists a measure ''μ'' supported on
'a'', ''b'' such that
:
for every ''ƒ'' ∈ ''C''
0(
'a'', ''b''.
Thus the existence of the measure
is equivalent to (). Using a representation theorem for positive polynomials on
'a'', ''b'' one can reformulate () as a condition on
Hankel matrices.
See and for more details.
Uniqueness (or determinacy)
The uniqueness of ''μ'' in the Hausdorff moment problem follows from the
Weierstrass approximation theorem
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
, which states that
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
s are
dense
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
under the
uniform norm
In mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions defined on a set the non-negative number
:\, f\, _\infty = \, f\, _ = \sup\left\.
This norm is also called the , the , the , or, when th ...
in the space of
continuous functions
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
on
, 1 For the problem on an infinite interval, uniqueness is a more delicate question; see
Carleman's condition In mathematics, particularly, in analysis, Carleman's condition gives a sufficient condition for the determinacy of the moment problem. That is, if a measure \mu satisfies Carleman's condition, there is no other measure \nu having the same moment ...
,
Krein's condition and . There are distributions, such as
log-normal distribution
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is log-normally distributed, then has a norma ...
s, which have finite moments for all the positive integers but where other distributions have the same moments.
Variations
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
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
. See also:
Chebyshev–Markov–Stieltjes inequalities and .
See also
*
Stieltjes moment problem
*
Hamburger moment problem In mathematics, the Hamburger moment problem, named after Hans Ludwig Hamburger, is formulated as follows: given a sequence (''m''0, ''m''1, ''m''2, ...), does there exist a positive Borel measure ''μ'' (for instance, the measure determined by t ...
*
Hausdorff moment problem In mathematics, the Hausdorff moment problem, named after Felix Hausdorff, asks for necessary and sufficient conditions that a given sequence be the sequence of moments
:m_n = \int_0^1 x^n\,d\mu(x)
of some Borel measure supported on the clo ...
*
Moment (mathematics)
*
Carleman's condition In mathematics, particularly, in analysis, Carleman's condition gives a sufficient condition for the determinacy of the moment problem. That is, if a measure \mu satisfies Carleman's condition, there is no other measure \nu having the same moment ...
*
Hankel matrix In linear algebra, a Hankel matrix (or catalecticant matrix), named after Hermann Hankel, is a square matrix in which each ascending skew-diagonal from left to right is constant, e.g.:
\qquad\begin
a & b & c & d & e \\
b & c & d & e & f \\
c & d & ...
References
*
* (translated from the Russian by N. Kemmer)
* (Translated from the Russian by D. Louvish)
*{{cite book , last1 = Schmüdgen , first1 = Konrad , title = The moment problem , publisher = Springer International Publishing , year = 2017
Mathematical analysis
Hilbert space
Probability problems
Moment (mathematics)
Mathematical problems
Real algebraic geometry
Optimization in vector spaces