Moment matrix
A matrix containing the moments of a probability distribution (cf. also Moment; Moments, method of (in probability theory)). For example, if is a probability distribution on a set I \subset \mathbf{C}, then m _ { k } = \int _ { I } x ^ { k } d \psi ( x ) is its kth 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) |
Moment matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Moment_matrix&oldid=51034