2 - Small Phenomena, Big Implications

Summary

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.