Stochastic LU factorizations, Darboux transformations and urn models

TL;DR

This paper explores stochastic LU and UL factorizations of transition matrices for certain random walks, linking them to Darboux transformations and urn models, with specific conditions and spectral measure identifications.

Contribution

It introduces conditions for stochastic LU and UL factorizations using continued fractions, and connects these to Darboux transformations and urn models for specific random walks.

Findings

Established conditions for stochastic LU and UL factorizations.

Linked Darboux transformations to spectral measure analysis.

Provided urn models for the studied random walks.

Abstract

We consider UL (and LU) decompositions of the one-step transition probability matrix of a random walk with state space the nonnegative integers, with the condition that both upper and lower triangular matrices in the factorization are also stochastic matrices. We give conditions on the free parameter of the UL factorization in terms of certain continued fraction such that this stochastic factorization is possible. By inverting the order of multiplication (also known as a Darboux transformation) we get a new family of random walks where it is possible the identify the spectral measures in terms of a Geronimus transformation. The same can be done for the LU factorization but now 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 give urn models associated with all the random walks in question.