Moment problem

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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

m_n = \int_{-\infty}^\infty x^n \,d\mu(x)\,.\,\!

More generally, one may consider

m_n = \int_{-\infty}^\infty M_n(x) \,d\mu(x)\,.\,\!

for an arbitrary sequence of functions Mn.

Introduction

In the classical setting, μ is a measure on the real line, and M is the sequence { xn : n = 0, 1, 2, ... }. 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 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 without loss of generality may be taken as [0, 1].

Existence

A sequence of numbers mn is the sequence of moments of a measure μ if and only if a certain positivity condition is fulfilled; namely, the Hankel matrices Hn,

(H_n)_{ij} = m_{i+j}\,,\,\!

should be positive semi-definite.[clarification needed] A condition of similar form is necessary and sufficient for the existence of a measure \mu supported on a given interval [ab].

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

P(x) = \sum_k a_k x^k \,\!

to

\sum_k a_k m_k.\,\!

If mkn are the moments of some measure μ supported on [ab], then evidently

φ(P) ≥ 0 for any polynomial P that is non-negative on [ab].

 

 

 

 

(1)

Vice versa, if (1) holds, one can apply the M. Riesz extension theorem and extend \phi to a functional on the space of continuous functions with compact support C0([ab]), so that

\qquad \varphi(f) \ge 0\text{ for any } f \in C_0([a,b])

 

 

 

 

(2)

such that ƒ ≥ 0 on [ab].

By the Riesz representation theorem, (2) holds iff there exists a measure μ supported on [ab], such that

 \phi(f) = \int f \, d\mu\,\!

for every ƒ ∈ C0([ab]).

Thus the existence of the measure \mu is equivalent to (1). Using a representation theorem for positive polynomials on [ab], one can reformulate (1) as a condition on Hankel matrices.

See Shohat & Tamarkin 1943 and Krein & Nudelman 1977 for more details.

Uniqueness (or determinacy)

The uniqueness of μ 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; see Carleman's condition, Krein's condition and Akhiezer (1965).

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. See also: Chebyshev–Markov–Stieltjes inequalities and Krein & Nudelman 1977.

See also

References

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