4 - Irreducible and aperiodic Markov chains

Summary

For several of the most interesting results in Markov theory, we need to put certain assumptions on the Markov chains we are considering. It is an important task, in Markov theory just as in all other branches of mathematics, to find conditions that on the one hand are strong enough to have useful consequences, but on the other hand are weak enough to hold (and be easy to check) for many interesting examples. In this chapter, we will discuss two such conditions on Markov chains: irreducibility and aperiodicity. These conditions are of central importance in Markov theory, and in particular they play a key role in the study of stationary distributions, which is the topic of Chapter 5. We shall, for simplicity, discuss these notions in the setting of homogeneous Markov chains, although they do have natural extensions to the more general setting of inhomogeneous Markov chains.

We begin with irreducibility, which, loosely speaking, is the property that “all states of the Markov chain can be reached from all others”. To make this more precise, consider a Markov chain (X₀, X₁, …) with state space S = {s₁, …, s_k} and transition matrix P. We say that a state s_icommunicates with another state s_j, writing s_i → s_j, if the chain has positive probability of ever reaching s_j when we start from s_i.