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

TL;DR

This paper develops stochastic Darboux transformations for quasi-birth-and-death processes, expanding the class of factorizations and applications, including new spectral measures and urn models.

Contribution

It introduces a generalized framework for stochastic factorizations of quasi-birth-and-death processes, enabling new spectral analysis and applications.

Findings

Many new factorizations due to block structure

Identification of spectral measures via matrix transformations

Development of urn models for specific cases

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.