This chapter is devoted to the spectral analysis of discrete-time birth–death chains on nonnegative integers, which are the most basic and important discrete-time Markov chains. These chains are characterized by a tridiagonal one-step transition probability matrix. The so-called Karlin–McGregor integral representation formula of the n-step transition probability matrix is obtained in terms of orthogonal polynomials with respect to a probability measure with support inside the interval [–1, 1]. An extensive collection of examples related to orthogonal polynomials is provided, including gambler’s ruin, the Ehrenfest model, the Bernoulli–Laplace model and the Jacobi urn model. The chapter concludes with applications of the Karlin–McGregor formula to probabilistic aspects of discrete-time birth–death chains, such as recurrence, absorption, the strong ratio limit property and the limiting conditional distribution. Finally, spectral methods are applied to discrete-time birth–death chains on the integers, which are not so much studied in the literature.

This chapter is devoted to the spectral analysis of birth–death processes on nonnegative integers, which are the most basic and important continuous-time Markov chains. These processes will be characterized by an infinitesimal operator which is a tridiagonal matrix whose spectrum is always contained in the negative real line (including 0). The Karlin–McGregor integral representation formula of the transition probability functions of the process is obtained in terms of orthogonal polynomials with respect to a probability measure with support inside a positive real interval. Although many of the results are similar or equivalent to those of discrete-time birth–death chains, the methods and techniques are quite different. The chapter gives an extensive collection of examples related to orthogonal polynomials, including the M/M/k queue for any k servers, the continuous-time Ehrenfest and Bernoulli–Laplace urn models, a genetics model of Moran and linear birth–death processes. As in the case of discrete-time birth–death chains, the Karlin–McGregor formula is applied to the probabilistic aspects of birth–death processes, such as processes with killing, recurrence, absorption, the strong ratio limit property, the limiting conditional distribution, the decay parameter, quasi-stationary distributions and bilateral birth–death processes on the integers.