Weak and Strong Interactions: Asymptotic Properties and Ramification

Abstract

Dealing with Markov chains having weak and strong interactions, Chapter 6 contains the formulations of three basic models, namely, the chains having recurrent states, the inclusion of absorbing states, and the inclusion of transient states. We have demonstrated that with some modifications, the approach for treating chains with recurrent states can be extended to the other two cases. In view of Remark 6.15 and Remark 6.20, the Markov chain with recurrent states is the most illustrative and representative one. As a result, in the remaining chapters, we only treat problems associated with this model, since such a model brings out the salient features of the underlying problems, and the techniques to be used for other cases will be similar. For brevity, unless otherwise noted, in the rest of the book, whenever the phrase “weak and strong interaction” is used, it refers to the case of singularly perturbed Markov chains with recurrent states. Similar approaches can be used for the other cases as well. Compared with Chapter 6, this chapter concentrates on exploiting detailed structures of the weak and strong interactions. In addition, it deals with convergence of the probability distribution with merely measurable generators.