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

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:478

Article

Grunbaum, F. Alberto¹  de la Iglesia, Manuel D.² 

[1] Univ Calif Berkeley, Dept Math, Berkeley, CA 94720 USA

[2] Univ Nacl Autonoma Mexico, Inst Matemat, Mexico City 04510, DF, Mexico

关键词: Quasi-birth-and-death processes;    LU block factorizations;    Darboux transformations;    Matrix-valued orthogonal polynomials;    Urn models;   

DOI  :  10.1016/j.jmaa.2019.05.048

来源: Elsevier

【 摘 要 】

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. (C) 2019 Elsevier Inc. All rights reserved.

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