Stochastic Darboux transformations for quasi-birth-and-death processes and urn models

We consider stochastic UL and LU block factorizations of the one-step transition probability matrix for a discrete-time quasi-birth-and-death process, namely a stochastic block tridiagonal matrix. The simpler case of random walks with only nearest neighbors transitions gives a unique LU factorizatio...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2019-10, Vol.478 (2), p.634-654
Main Authors: Grünbaum, F. Alberto, de la Iglesia, Manuel D.
Format: Article
Language:English
Subjects:
Darboux transformations
LU block factorizations
Matrix-valued orthogonal polynomials
Quasi-birth-and-death processes
Urn models
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Summary:We consider stochastic UL and LU block factorizations of the one-step transition probability matrix for a discrete-time quasi-birth-and-death process, namely a stochastic block tridiagonal matrix. The simpler case of random walks with only nearest neighbors transitions gives a unique LU factorization and a one-parameter family of factorizations in the UL case. The block structure considered here yields many more possible factorizations resulting in a much enlarged class of potential applications. By reversing the order of the factors (also known as a Darboux transformation) we get new families of quasi-birth-and-death processes where it is possible to identify the matrix-valued spectral measures in terms of a Geronimus (UL) or a Christoffel (LU) transformation of the original one. We apply our results to one example going with matrix-valued Jacobi polynomials arising in group representation theory. We also give urn models for some particular cases.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2019.05.048