Uniform and V-Geometric Ergodicity by Operator Methods
In this chapter, we will obtain new characterizations and proofs of the uniform ergodicity properties established in Chapter 15. We will consider a Markov kernel P as a linear operator on a set of probability measures endowed with a certain metric. An invariant probability measure is a fixed point of this operator, and therefore, a natural idea is to use a fixed-point theorem to prove convergence of the iterates of the kernel to the invariant distribution. To do so, in Section 18.1, we will first state and prove a version of the fixed-point theorem that suits our purposes. As appears in the fixed-point theorem, the main restriction of this method is that it can provide only geometric rates of convergence. These techniques will be again applied in Chapter 20, where we will be dealing with other metrics on the space of probability measures.