Matrix measures and random walks with a block tridiagonal transition matrix. (English) Zbl 1133.60339
Summary: We study the connection between matrix measures and random walks with a block tridiagonal transition matrix. We derive sufficient conditions such that the blocks of the \(n\)-step block tridiagonal 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.
MSC:
60J10 | Markov chains (discrete-time Markov processes on discrete state spaces) |
42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |