Preface

Summary

The basic object of study in this book is the theory of discrete-time Markov processes or, briefly, Markov chains, defined on a general measurable space and having stationary transition probabilities.

The theory of Markov chains with values in a countable set (discrete Markov chains) can nowadays be regarded as part of classical probability theory. Its mathematical elegance, often involving the use of simple probabilistic arguments, and its practical applicability have made discrete Markov chains standard material in textbooks on probability theory and stochastic processes.

It is clear that the analysis of Markov chains on a general state space requires more elaborate techniques than in the discrete case. Despite these difficulties, by the beginning of the 1970s the general theory had developed to a mature state where all of the fundamental problems – such as cyclicity, the recurrence-transience classification, the existence of invariant measures, the convergence of the transition probabilities – had been answered in a satisfactory manner. At that time also several monographs on general Markov chains were published (e.g. Foguel, 1969 a; Orey, 1971; Rosenblatt, 1971; Revuz, 1975).

The primary motivation for writing this book has been in the recent developments in the theory of general (irreducible) Markov chains. In particular, owing to the discovery of embedded renewal processes, the ‘elementary’ techniques and. constructions based on the notion of regeneration, and common in the study of discrete chains, can now be applied in the general case.