In this chapter we introduce the culminating form of the geometric ergodicity theorem, and show that such convergence can be viewed as geometric convergence of an operator norm; simultaneously, we show that the classical concept of uniform (or strong) ergodicity, where the convergence in (13.4) is bounded independently of the starting point, becomes a special case of this operator norm convergence.

We also take up a number of other consequences of the geometric ergodicity properties proven in Chapter 15, and give a range of examples of this behavior. For a number of models, including random walk, time series and state space models of many kinds, these examples have been held back to this point precisely because the strong form of ergodicity we now make available is met as the norm, rather than as the exception. This is apparent in many of the calculations where we verified the ergodic drift conditions (V2) or (V3): often we showed in these verifications that the stronger form (V4) actually held, so that unwittingly we had proved V-uniform or geometric ergodicity when we merely looked for conditions for ergodicity.