Birth-death chains on a spider: spectral analysis and reflecting-absorbing factorization

TL;DR

This paper analyzes birth-death chains on a spider graph using spectral methods, providing explicit transition probabilities, factorization techniques, and transformations to generate new related chains, with applications to random walks.

Contribution

It introduces spectral analysis and factorization methods for birth-death chains on a spider, including matrix-valued orthogonal polynomials and Darboux transformations, extending existing stochastic models.

Findings

Explicit n-step transition probabilities via matrix-valued orthogonal polynomials.

Conditions for reflecting-absorbing factorization of the transition matrix.

Construction of new birth-death chains through Darboux transformations.

Abstract

We consider discrete-time birth-death chains on a spider, i.e. a graph consisting of $N$ discrete half lines on the plane that are joined at the origin. This process can be identified with a discrete-time quasi-birth-death process on the state space $\mathbb{N}_0 \times \{1, 2, \dots, N\},$ represented by a block tridiagonal transition probability matrix. We prove that we can analyze this process by using spectral methods and obtain the $n$-step transition probabilities in terms of a weight matrix and the corresponding matrix-valued orthogonal polynomials (the so-called Karlin-McGregor formula). We also study under what conditions we can get a reflecting-absorbing factorization of the birth-death chain on a spider which can be seen as a stochastic UL block factorization of the transition probability matrix of the quasi-birth-death process. With this factorization we can perform a discrete Darboux transformation and get new families of "almost" birth-death chains on a spider. The spectral matrix associated with the Darboux transformation will be a Geronimus transformation of the original spectral matrix. Finally, we apply our results to the random walk on a spider, i.e. with constant transition probabilities.