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Stochastic LU factorizations, Darboux transformations and urn models

Part of: Markov processes Nontrigonometric harmonic analysis

Published online by Cambridge University Press:  16 November 2018

F. Alberto Grünbaum  and
Manuel D. de la Iglesia
F. Alberto Grünbaum*
Affiliation:
University of California, Berkeley
Manuel D. de la Iglesia*
Affiliation:
Universidad Nacional Autónoma de México
*
* Postal address: Department of Mathematics, University of California, Berkeley, CA 94720, USA.
** Postal address: Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, C.U., 04510, Ciudad de México, México. Email address: mdi29@im.unam.mx
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Abstract

We consider upper‒lower (UL) (and lower‒upper (LU)) factorizations of the one-step transition probability matrix of a random walk with the state space of nonnegative integers, with the condition that both upper and lower triangular matrices in the factorization are also stochastic matrices. We provide conditions on the free parameter of the UL factorization in terms of certain continued fractions such that this stochastic factorization is possible. By inverting the order of the factors (also known as a Darboux transformation) we obtain a new family of random walks where it is possible to state the spectral measures in terms of a Geronimus transformation. We repeat this for the LU factorization but without a free parameter. Finally, we apply our results in two examples; the random walk with constant transition probabilities, and the random walk generated by the Jacobi orthogonal polynomials. In both situations we obtain urn models associated with all the random walks in question.

Keywords

Darboux transformationLU factorizationorthogonal polynomialrandom walkurn model

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 42C05: Orthogonal functions and polynomials, general theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 3 , September 2018 , pp. 862 - 886
Copyright
Copyright © Applied Probability Trust 2018 

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