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

F. Alberto Grunbaum; Manuel D. de la Iglesia

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arXiv · 1802.05221

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

F. Alberto Grunbaum·Manuel D. de la Iglesia

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Abstract

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.

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F. Alberto Grunbaum, Manuel D. de la Iglesia. 2018-02-14. Stochastic Darboux transformations for quasi-birth-and-death processes and urn models. https://arxiv.org/abs/1802.05221

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