Coupling Methods for Markov Chains and the Renewal Theorem for Lattice Distributions

Abstract

The coupling method is a powerful tool of stochastic analysis that has enjoyed many successes since its original introduction by Doeblin (Revue Math de l’Union Interbalkanique 2:77–105, 1938) to prove convergence to a unique invariant probability for finite state Markov chains. In fact it was applied in Chapter 5 to obtain an error bound in the Poisson approximation to the binomial distribution, i.e., the law of rare events. The convergence to steady state for a class of finite state Markov chains together with a proof of a related powerful result, the renewal theorem, is presented. In the latter one seeks to find how much time a general random walk on the integers with increasing paths, i.e., having non-negative integer-valued displacements, spends in an interval of length m, say (n, n + m]. Renewal theory computes the precise amount asymptotically as n →∞. This chapter is devoted to a cornerstone theorem in the case of integer-valued renewal times, while the much more general theory is provided as a special topic in Chapter 25.