Topics in the theory and applications of Markov chains

Abstract

The dissertation which follows is concerned with various aspects of behaviour within a set of states, J, of a discrete-time Markov chain, {Xn }, on a denumerable state space, S. A basic assumption with regard to J is that escape from any state of J into S-J may occur in a finite number of steps with positive probability. Since we are concerned only with behaviour within J, we may in general take J = {1,2,3,...} and represent S-J as a single absorbing state {0}. Thus without loss of generality S = {0,1,2,,..} with the states 1,2,3,... being transient. In addition, we frequently assume in the sequel that J is a single irreducible (i.e. intercommunicating) class, and sometimes that this class is aperiodic, these assumptions corresponding to the situations of greatest theoretical and practical importance. The subject matter which we treat falls naturally into two parts, according to which the thesis is divided. The aim of Part One is to develop results and techniques applicable to a wide class of problems, under general assumptions. This is done in the first three chapters, in which specific chains enter only as examples. On the other hand, specialized techniques are often applicable to specific chains of wide interest, such as certain models in genetics. This is particularly true of the Galton-Watson process, which is the subject of the following three chapters (Part Two) of the thesis. Three distinct but related aspects of transient behaviour within J are studied in the first part, each corresponding to a chapter.