14 - Further reading

Summary

Markov theory is a huge subject (much bigger than indicated by these notes), and consequently there are many books written on it. Three books that have influenced the present text are the ones by Brémaud [B], Grimmett & Stirzaker [GS], and (the somewhat more advanced book by) Durrett [Du]. Another nice introduction to the topic is the book by Norris [N]. Some of my Swedish compatriots will perhaps prefer to consult the texts by Rydén & Lindgren [RL] and Enger & Grandell [EG]. The reader can find plenty of additional material (more general theory, as well as other directions for applications) in any of these references.

Still on the Markov theory side (Chapters 2–6) of this text, there are two particular topics that I would warmly recommend for further study to anyone with a taste for mathematical elegance and the power and simplicity of probabilistic arguments: The first one is the coupling method, which was used to prove Theorems 5.2 and 8.1, and which also underlies the algorithms in Chapters 10–12; see the books by Lindvall [L] and by Thorisson [T]. The second topic is the relation between reversible Markov chains and electrical networks, which is delightfully treated in the book by Doyle & Snell [DSn]. Häggström [H] gives a short introduction in Swedish.

Another goldmine for the ambitious student is the collection of papers edited by Snell [Sn], where many exciting topics in probability, several of which concern Markov chains and/or randomized algorithms, are presented on a level accessible to advanced undergraduates.