2018/02/14 by F. Alberto Grünbaum, Grunbaum, F. Alberto, Manuel D. de la Iglesia +1
Mathematics · Physics and Astronomy · #33C45 #42C05 #60J10 #60J60 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Probability (math.PR) #Quantum chaos and dynamical systems #Quantum optics and atomic interactions #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.1802.05221
openalex publication_date 2018/02/14 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
We consider stochastic UL and LU block factorizations of the one-step\ntransition probability matrix for a discrete-time quasi-birth-and-death\nprocess, namely a stochastic block tridiagonal matrix. The simpler case of\nrandom walks with only nearest neighbors transitions gives a unique LU\nfactorization and a one-parameter family of factorizations in the UL case. The\nblock structure considered here yields many more possible factorizations\nresulting in a much enlarged class of potential applications. By reversing the\norder of the factors (also known as a Darboux transformation) we get new\nfamilies of quasi-birth-and-death processes where it is possible to identify\nthe matrix-valued spectral measures in terms of a Geronimus (UL) or a\nChristoffel (LU) transformation of the original one. We apply our results to\none example going with matrix-valued Jacobi polynomials arising in group\nrepresentation theory. We also give urn models for some particular cases.\n