Matrix Measures and Random Walks with a Block Tridiagonal Transition Matrix

Distributions on matrix moment spaces

“…Typical examples include the three-termrecursion, quadrature formulas and ratio asymptotics. Applications to stochastic processes with two-dimensional state space were discussed by Dette et al (2006), who expressed transition probabilities and the recurrence of states in terms of matrix measures and matrix orthogonal polynomials. In contrast to moment spaces corresponding to (probability) measures the structure of moment spaces corresponding to matrix measures is much richer and not very well understood.…”

Some bivariate stochastic models arising from group representation theory

“…In the last few years many progresses have been made in this direction. For discretetime quasi-birth-and-death processes the extension of the Karlin-McGregor formula was given independently in [6,9], while for continuous-time in [5]. For switching diffusion processes see [14].…”

Matrix polynomials orthogonal with respect to a non-symmetric matrix of measures

“…MOPs -which are known to satisfy a three-term recurrence relation of type (1.3) -are investigated for example in the theory of random walks on graphs or in the theory of multiple birth-and-death processes (cf. [5,8]), where such thre term formulae appear. Many properties of random walk or birth-and-death processes can be obtained from the matrix of measures of orthogonality.…”

“…The turn of millennia saw a flurry of articles, among them for instance [2,[4][5][6][7]11,13,14]. They consider only positive definite matrices of measures of orthogonality.…”