Moment matrix

A matrix containing the moments of a probability distribution (cf. also Moment; Moments, method of (in probability theory)). For example, if $\psi$ is a probability distribution on a set $I \subset \mathbf{C}$, then $m _ { k } = \int _ { I } x ^ { k } d \psi ( x )$ is its $k$th order moment. If $\psi$ and thus the moments are given, then a linear functional $L$ is defined on the set of polynomials by $L ( x ^ { k } ) = m _ { k }$, $k = 0,1 , \ldots$. The inverse problem is called a moment problem (cf. also Moment problem): Given the sequence of moments $m_k$, $k = 0,1 , \ldots$, find the necessary and sufficient conditions for the existence of and an expression for a positive distribution (a non-decreasing function with possibly infinitely many points of increase) that gives the integral representation of that linear functional. A positive distribution can only exist if $L ( p ) > 0$ for any polynomial $p$ that is positive on $I$.

For the Hamburger moment problem (cf. also Complex moment problem, truncated), $I$ is the real axis and the polynomials are real, so the functional $L$ is positive if $L ( p ^ { 2 } ( x ) ) > 0$ for any non-zero polynomial $p$ and this implies that the moment matrices, i.e., the Hankel matrices of the moment sequence, $M _ { n } = [ m _ { i+j }] _ { i , j = 0 } ^ { n }$, are positive definite for all $n = 0,1 , \dots$ (cf. also Hankel matrix). This is a necessary and sufficient condition for the existence of a solution.

For the trigonometric moment problem, $I$ is the unit circle in the complex plane and the polynomials are complex, so that "positive definite" here means that $L ( | p ( z ) | ^ { 2 } ) > 0$ for all non-zero polynomials $p$. The linear functional is automatically defined on the space of Laurent polynomials (cf. also Laurent series) since $m _ { - k } = L ( z ^ { - k } ) = \overline { L ( z ^ { k } ) } = \overline { m } _ { k }$. Positive definite now corresponds to the Toeplitz moment matrices $M _ { n } = [ m _ { i - j} ] _ { i ,\, j = 0 } ^ { n }$ being positive definite for all $n = 0,1,2 , \dots$ (cf. also Toeplitz matrix). Again this is the necessary and sufficient condition for the existence of a (unique) solution to the moment problem.

Once the positive-definite linear functional is given, one can define an inner product on the space of polynomials as $\langle f , g \rangle = L ( f ( x ) g ( x ) )$ in the real case or as $\langle f , g \rangle = L ( f ( z ) \overline { g ( z ) } )$ in the complex case. The moment matrix is then the Gram matrix for the standard basis $m _ { i + j} = \langle x ^ { i } , x ^ { j } \rangle$ or $m _ { i -j } = \langle x ^ { i } , x ^ { j } \rangle$.

Generalized moments correspond to the use of non-standard basis functions for the polynomials or for possibly other spaces. Consider a set of basis functions $f _ { 0 } , f _ { 1 } , \dots$ that span the space $\mathcal{L}$. The modified or generalized moments are then given by $m _ { k } = L ( f _ { k } )$. The moment problem is to find a positive distribution function $\psi$ that gives an integral representation of the linear functional on $\mathcal{L}$. However, to define an inner product, one needs the functional to be defined on $\mathcal{R} = \mathcal{L}. \mathcal{L}$ (in the real case) or on $\mathcal{R} = \mathcal{L}. \overline { \mathcal{L} }$ (in the complex case). This requires a doubly indexed sequence of "moments" $m _ { i j } = \langle f _ { i } , f _ { j } \rangle$. Finding a distribution for an integral representation of $L$ on $\mathcal{R}$ is called a strong moment problem.

The solution of moment problems is often obtained using an orthogonal basis. If the $f _ { k }$ are orthonormalized to give the functions $\phi _ { 0 } , \phi _ { 1 } , \ldots$, then the moment matrix $M _ { n } = [ m _ { i j } ] _ { i , j = 0 } ^ { n }$ can be used to give explicit expressions; namely $\phi _ { n } ( z ) = M _ { n } ( z ) / \sqrt { \mathcal{M} _ { n - 1} \mathcal{M} _ { n }} $ where $\mathcal{M} _ { - 1 } = 0$, $\mathcal{M} _ { 0 } ( z ) = f _ { 0 } ( z )$ and for $n \geq 1$, ${\cal M} _ { n } = \operatorname { det } M _ { n }$ with

\begin{equation*} M _ { n } ( z ) = \left( \begin{array} { c c c } { \langle f _ { 0 } , f _ { 0 } \rangle } & { \dots } & { \langle f _ { 0 } , f _ { n } \rangle } \\ { \vdots } & { \square } & { \vdots } \\ { \langle f _ { n - 1 } , f _ { 0 } \rangle } & { \dots } & { \langle f _ { n - 1 } , f _ { n } \rangle } \\ { f _ { 0 } ( z ) } & { \dots } & { f _ { n } ( z ) } \end{array} \right). \end{equation*}

The leading coefficient in the expansion $\phi _ { n } ( z ) = \kappa _ { n } f _ { n } ( z ) +\dots $ satisfies $| \kappa _ { n } | ^ { 2 } = {\cal M} _ { n - 1 } / {\cal M} _ { n }$.

References

[a1]	N.I. Akhiezer, "The classical moment problem" , Oliver & Boyd (1969) (In Russian)
[a2]	J.A. Shohat, J.D. Tamarkin, "The problem of moments" , Math. Surveys , 1 , Amer. Math. Soc. (1943) (In Russian)