An note on the maximization of matrix valued Hankel determinants with application

In this note we consider the problem of maximizing the determinant of moment matrices of matrix measures. The maximizing matrix measure can be characterized explicitly by having equal (matrix valued) weights at the zeros of classical (one dimensional) orthogonal polynomials. The results generalize classical work of Schoenberg (1959) to the case of matrix measures. As a statistical application we consider several optimal design problems in linear models, which generalize the classical weighing design problems. --Matrix measures,Hankel matrix,orthogonal polynomials,approximate optimal designs,spring balance weighing designs

Matrix measures and random walks

In this paper we study the connection between matrix measures and random walks with a tridiagonal block transition matrix. We derive sufficient conditions such that the blocks of the n-step transition matrix of the Markov chain can be represented as integrals with respect to a matrix valued spectral measure. Several stochastic properties of the processes are characterized by means of this matrix measure. In many cases this measure is supported in the interval [-1, 1]. The results are illustrated by several examples including random walks on a grid and the embedded chain of a queuing system. --Markov chain,block tridiagonal transition matrix,spectral measure,matrix measure,quasi birth and death processes,canonical moments

Optimum designs when the observations are second-order processes

Let the process {Y(x,t) : t [epsilon] T} be observable for each x in some compact set X. Assume that Y(x, t) = [theta]0f0(x)(t) + ... + [theta]kfk(x)(t) + N(t) where fi are continuous functions from X into the reproducing kernel Hilbert space H of the mean zero random process N. The optimum designs are characterized by an Elfving's theorem with the closed convex hull of the set {([phi], f(x))H : ||[phi] ||HOptimum design estimating a linear form stochastic process reproducing kernel Hilbert space extreme points Elfving's theorem

Matrix measures and random walks

In this paper we study the connection between matrix measures and random walks with a tridiagonal block transition matrix. We derive sufficient conditions such that the blocks of the n-step transition matrix of the Markov chain can be represented as integrals with respect to a matrix valued spectral measure. Several stochastic properties of the processes are characterized by means of this matrix measure. In many cases this measure is supported in the interval [−1, 1]. The results are illustrated by several examples including random walks on a grid and the embedded chain of a queuing system