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

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.