Hamburger moment problem

In mathematics, the Hamburger moment problem, named after Hans Ludwig Hamburger, is formulated as follows: given a sequence Template:Math, does there exist a positive Borel measure Template:Mvar (for instance, the measure determined by the cumulative distribution function of a random variable) on the real line such that

$m_n=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{x}^{n}d\mu (x)$?

In other words, an affirmative answer to the problem means that Template:Math is the sequence of moments of some positive Borel measure Template:Mvar.

The Stieltjes moment problem, Vorobyev moment problem, and the Hausdorff moment problem are similar but replace the real line by $[0,+\mathrm{\infty })$ (Stieltjes and Vorobyev; but Vorobyev formulates the problem in the terms of matrix theory), or a bounded interval (Hausdorff).

Characterization

The Hamburger moment problem is solvable (that is, Template:Math is a sequence of moments) if and only if the corresponding Hankel kernel on the nonnegative integers

$A=\left(\begin{array}{cccc}{m}_0& m_1& m_2& \cdots \\ m_1& m_2& m_3& \cdots \\ m_2& m_3& m_4& \cdots \\ ⋮& ⋮& ⋮& \ddots \end{array}\right)$

is positive definite, i.e.,

$\sum _{j,k\ge 0}{m}_{j+k}{c}_j\overline{c_k}\ge 0$

for every arbitrary sequence Template:Math of complex numbers that are finitary (i.e., Template:Math except for finitely many values of Template:Mvar).

For the "only if" part of the claims simply note that

$\sum _{j,k\ge 0}{m}_{j+k}{c}_j\overline{c_k}=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{\left|\sum _{j\ge 0}{c}_j{x}^j\right|}^{2}d\mu (x)$,

which is non-negative if $\mu$ is non-negative.

We sketch an argument for the converse. Let Template:Math be the nonnegative integers and Template:Math denote the family of complex valued sequences with finitary support. The positive Hankel kernel Template:Mvar induces a (possibly degenerate) sesquilinear product on the family of complex-valued sequences with finite support. This in turn gives a Hilbert space

$(\mathscr{H},⟨,⟩)$

whose typical element is an equivalence class denoted by Template:Math.

Let Template:Mvar be the element in Template:Math defined by Template:Math. One notices that

$⟨[e_{n+1}],[e_m]⟩=A_{m,n+1}=m_{m+n+1}=⟨[e_n],[e_{m+1}]⟩$.

Therefore, the shift operator Template:Mvar on $\mathscr{H}$, with Template:Math, is symmetric.

On the other hand, the desired expression

$m_n=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{x}^{n}d\mu (x)$

suggests that Template:Mvar is the spectral measure of a self-adjoint operator. (More precisely stated, Template:Mvar is the spectral measure for an operator $\bar{T}$ defined below and the vector [1], Template:Harv). If we can find a "function model" such that the symmetric operator Template:Mvar is [[multiplication operator|multiplication by Template:Mvar]], then the spectral resolution of a self-adjoint extension of Template:Mvar proves the claim.

A function model is given by the natural isomorphism from F₀(Z⁺) to the family of polynomials, in one single real variable and complex coefficients: for n ≥ 0, identify e_n with xⁿ. In the model, the operator T is multiplication by x and a densely defined symmetric operator. It can be shown that T always has self-adjoint extensions. Let $\bar{T}$ be one of them and μ be its spectral measure. So

$⟨\stackrel{n}{\stackrel{‾}{T}}[1],[1]⟩=\int x^{n}d\mu (x)$.

On the other hand,

$⟨\stackrel{n}{\stackrel{‾}{T}}[1],[1]⟩=⟨T^n[e_0],[e_0]⟩=m_n$.

For an alternative proof of the existence that only uses Stieltjes integrals, see also,Template:Sfn in particular theorem 3.2.

Uniqueness of solutions

The solutions form a convex set, so the problem has either infinitely many solutions or a unique solution.

Consider the Template:Math Hankel matrix

${\mathrm{\Delta }}_n=\left[\begin{array}{ccccc}{m}_0& m_1& m_2& \cdots & m_n\\ m_1& m_2& m_3& \cdots & m_{n+1}\\ m_2& m_3& m_4& \cdots & m_{n+2}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ m_n& m_{n+1}& m_{n+2}& \cdots & m_{2n}\end{array}\right]$.

Positivity of Template:Mvar means that, for each Template:Mvar, Template:Math. If Template:Math, for some Template:Mvar, then

$(\mathscr{H},⟨,⟩)$

is finite-dimensional and Template:Mvar is self-adjoint. So in this case the solution to the Hamburger moment problem is unique and Template:Mvar, being the spectral measure of Template:Mvar, has finite support.

More generally, the solution is unique if there are constants Template:Mvar and Template:Mvar such that, for all Template:Mvar, Template:Math Template:Harv. This follows from the more general Carleman's condition.

There are examples where the solution is not unique; see e.g.Template:Sfn

Polynomials

The Hamburger moment problem is intimately related to orthogonal polynomials on the real line. That is, assume $\{m_n{\}}_{n\in {\mathbb{N}}_0}$ is the moment sequence of some positive measure $\mu$ on $ℝ$. Then for any polynomial $$p(x)=\sum _{j=0}^n{a}_j{x}^j\in ℝ,$$ it holds that $$\int p(x{)}^{2}d\mu =\int \left(\sum _{j,k=0}^n{a}_j{a}_k{x}^{j+k}\right)d\mu =\sum _{j,k=0}^n{a}_j{a}_k{m}_{j+k}\ge 0,\mathrm{\forall }n\in {\mathbb{N}}_0,$$ such that the Hankel matrix is positive semidefinite. This is a necessary condition for a sequence to be a moment sequence and a sufficient condition for the existence of a positive measure.Template:Sfn

The Gram–Schmidt procedure gives a basis of orthogonal polynomials in which the operator: $\bar{T}$ has a tridiagonal Jacobi matrix representation. This in turn leads to a tridiagonal model of positive Hankel kernels.

An explicit calculation of the Cayley transform of Template:Mvar shows the connection with what is called the Nevanlinna class of analytic functions on the left half plane. Passing to the non-commutative setting, this motivates Krein's formula which parametrizes the extensions of partial isometries.

The cumulative distribution function and the probability density function can often be found by applying the inverse Laplace transform to the moment generating function

$m(t)=\sum _{n=0}{m}_n\frac{t^n}{n!}$,

provided that this function converges.

References

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