Quasi-Birth-and-Death Processes and Matrix-Valued Orthogonal Polynomials

We consider a matrix-valued spectral decomposition of a family of block-tridiagonal matrices arising as the transition matrices of so-called quasi-birth-and-death processes. This representation is a generalization of the Karlin-McGregor representation for the $n$-step transition probabilities of a birth-and-death process via a system of orthogonal polynomials. At the heart of the representation is a self-adjoint matrix-valued measure associated to the process. We make use of a previously known formula relating the Stieltjes transform of this measure to that of the measure associated to the “0th associated process,” generalizing a theorem of Karlin and McGregor, to compute the Stieltjes transform of the spectral measure for several examples. In addition, we apply matrix-valued orthogonal polynomial techniques to the study of “sin-graphs” and higher-dimensional birth-and-death processes, for which the relevant polynomials are multivariate.

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