Despite our best efforts, we understand that the scope of this book inevitably leads to the potential for confusion in readers new to the subject, especially in view of the variety of approaches to stability which we have given, the many related and perhaps (until frequently used) forgettable versions of the “Foster-Lyapunov” drift criteria, and the sometimes widely separated conditions on the various models which are introduced throughout the book.

At the risk of repetition, we therefore gather together in this Appendix several discussions which we hope will assist in giving both the big picture, and a detailed illustration of how the structural results developed in this book may be applied in different contexts.

We first give a succinct series of equivalences between and implications of the various classifications we have defined, as a quick “mud map” to where we have been. In particular, this should help to differentiate between those stability conditions which are “almost” the same.

Secondly, we list together the drift conditions, in slightly abbreviated form, together with references to their introduction and the key theorems which prove that they are indeed criteria for different forms of stability and instability. As a guide to their usage we then review the analysis of one specific model (the scalar threshold autoregression, or SETAR model).

This model incorporates a number of sub-models (specifically, random walks and scalar linear models) which we have already analyzed individually: thus, although not the most complex model available, the SETAR model serves to illustrate many of the technical steps needed to convert elegant theory into practical use in a number of fields of application.