Moment problem

In mathematics, a moment problem arises as the result of trying to invert the mapping that takes a measure $\mu$ to the sequence 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 $M_{n}$ .

Introduction[edit]

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

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 $μ$ is the complex unit circle instead of the real line.^[1]

Existence[edit]

A sequence of numbers $m_{n}$ is the sequence of moments of a measure $\mu$ if and only if a certain positivity condition is fulfilled; namely, the Hankel matrices $H_{n}$ ,

(H_{n})_{ij}=m_{i+j}\,,

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

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

P(x)=\sum _{k}a_{k}x^{k}

to

\sum _{k}a_{k}m_{k}.

If $m_{k}$ are the moments of some measure $\mu$ supported on $[a,b]$ , then evidently

\varphi (P)\geq 0

for any polynomial

P

that is non-negative on

[a,b]

.

(1)

Vice versa, if (1) holds, one can apply the M. Riesz extension theorem and extend $\varphi$ to a functional on the space of continuous functions with compact support $C_{c}([a,b])$ ), so that

\varphi (f)\geq 0

for any

f\in C_{c}([a,b]),\;f\geq 0.

(2)

By the Riesz representation theorem, (2) holds iff there exists a measure $\mu$ supported on $[a,b]$ , such that

\varphi (f)=\int f\,d\mu

for every $f\in C_{c}([a,b])$ .

Thus the existence of the measure $\mu$ is equivalent to (1). Using a representation theorem for positive polynomials on $[a,b]$ , one can reformulate (1) as a condition on Hankel matrices.^[2]^[3]

Uniqueness (or determinacy)[edit]

The uniqueness of $\mu$ 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.^[4] There are distributions, such as log-normal distributions, which have finite moments for all the positive integers but where other distributions have the same moments.

Formal solution[edit]

When the solution exists, it can be formally written using derivatives of the Dirac delta function as

d\mu (x)=\rho (x)dx,\quad \rho (x)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}\delta ^{(n)}(x)m_{n}

.

The expression can be derived by taking the inverse Fourier transform of its characteristic function.

Variations[edit]

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.^[3]

Probability[edit]

The moment problem has applications to probability theory. The following is commonly used:^[5]

Theorem (Fréchet-Shohat) — If ${\textstyle \mu }$ is a determinate measure (i.e. its moments determine it uniquely), and the measures ${\textstyle \mu _{n}}$ are such that

\forall k\geq 0\quad \lim _{n\rightarrow \infty }m_{k}\left[\mu _{n}\right]=m_{k}[\mu ],

then

{\textstyle \mu _{n}\rightarrow \mu }

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:

Corollary — If a sequence of probability distributions ${\textstyle \nu _{n}}$ satisfy

m_{2k}[\nu _{n}]\to {\frac {(2k)!}{2^{k}k!}};\quad m_{2k+1}[\nu _{n}]\to 0