Absorbing–reflecting factorizations for birth–death chains on the integers and their Darboux transformations

Abstract

We consider a new way of factorizing the transition probability matrix of a discrete-time birth–death chain on the integers by means of an absorbing and a reflecting birth–death chain to the state 0 and viceversa. First we will consider reflecting–absorbing factorizations of birth–death chains on the integers. We give conditions on the two free parameters such that each of the factors is a stochastic matrix. By inverting the order of the factors (also known as a Darboux transformation) we get new families of “almost” birth–death chains on the integers with the only difference that we have new probabilities going from the state 1 to the state −1 and viceversa. On the other hand an absorbing–reflecting factorization of birth–death chains on the integers is only possible if both factors are split into two separated birth–death chains at the state 0. Therefore it makes more sense to consider absorbing–reflecting factorizations of “almost” birth–death chains with extra transitions between the states 1 and −1 and with some conditions. This factorization is now unique and by inverting the order of the factors we get a birth–death chain on the integers. In both cases we identify the spectral matrices associated with the Darboux transformation, the first one being a Geronimus transformation and the second one a Christoffel transformation of the original spectral matrix. We also study the probabilistic implications of both transformations. Finally, we apply our results to examples of chains with constant transition probabilities.