moment problem
(statistics) The problem of finding a distribution whose moments have specified values, or of determining whether such a distribution exists.
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(statistics) The problem of finding a distribution whose moments have specified values, or of determining whether such a distribution exists.
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

where Mn(x) is the nth in a list of monomials, for n = 0, 1, 2, 3, ... .
In the classical setting, μ is a measure on the real line, and M is in the sequence { xn : n = 0, 1, 2, ... }, giving moments mn for n = 0, 1, 2, 3, ... . It is in this form that the question would appear in probability theory, of asking to whether there is a probability measure having specified mean, variance and so on.
There are three named classical moment problems: the Hamburger moment
problem in which the support of μ is allowed to be the whole real line; the
Stieltjes moment problem, for [0, +∞); and the Hausdorff moment problem for a bounded interval, which
It was realized that this is closely connected to Hilbert spaces and spectral theory. In more concrete terms, there is a condition on a positive measure μ, namely that

for every complex-valued polynomial P(x), unless P vanishes on the support of μ. This gives rise to matrix conditions, necessary on any sequence of moments, namely that certain Hankel matrices are positive semi-definite.
The uniqueness of μ in the Hausdorff moment problem follows because polynomials are dense in the uniform norm on [0, 1]. For the problem on an infinite interval, uniqueness is a more delicate question; see Carleman's condition, Krein's condition and Ref. 2.
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. See also: Chebyshev-Markov-Stieltjes inequalities and Ref. 3.
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