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Suppose one is interested in stochastic processes for the purpose getting a theoretical understanding of MCMC. They already have a decent understanding of probability theory (let's say at the level Billingsley) and want to develop enough tools to potentially prove theorems about the Markov Chains they construct at a research level.

What book or sequence of books would be reasonable for getting up to speed on the theory?

EDIT: This is not a question about applying MCMC. It is essentially asking for a mathematics textbook. Something less like Robert and Casella and more like Meyn and Tweedie (an answer I probably would have accepted when I asked this question).

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    $\begingroup$ There's been some other discussions on similar topics -- see e.g. Textbook deriving Metropolis-Hastings and Gibbs Sampling $\endgroup$ – StasK Nov 8 '12 at 1:03
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    $\begingroup$ Possible duplicate of Good sources for learning Markov chain Monte Carlo (MCMC) $\endgroup$ – Tim♦ Jul 26 '16 at 7:10
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    $\begingroup$ @Tim I disagree. Both the links ask the question of good books for MCMC sampling methods and theoretical understanding of MCMC methods. This question asks for books to understand the stochastic processes used in MCMC. The books in my answer (except for Brooks et al.) really don't talk about MCMC, but just general state space Markov chains. Which is I think what the OP is asking. $\endgroup$ – Greenparker Jul 26 '16 at 9:43
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    $\begingroup$ @Tim This is not a duplicate, as hinted at from my previous reply to StasK. When I asked this question (almost 4 years ago!) it was because I did not find the treatment of Robert and Casella rigorous enough. None of the texts, I think, in the post you linked to would have been adequate answers to this question. $\endgroup$ – guy Jul 26 '16 at 14:25
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    $\begingroup$ @Greenparker I was planning to, but since this question has generated more interest in the last day I thought I would wait to see if it generated any more answers. Marking as correct will stop others from answering. I'll accept an answer in probably the next day or so. $\endgroup$ – guy Jul 26 '16 at 15:55
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Since you are asking for MCMC theory, I am assuming Markov chains on general state space is of the most interest here. Here I provide some books/articles and what they are good for.

  1. Markov Chains and Stochastic Stability by Meyn and Tweedie. This is considered to be the most thorough book on the theory for Markov chains in MCMC. You will find that most research in the theory of MCMC refer to this book often. Their treatment is fairly measure theoretic, too much maybe for my taste. This is probably a good one stop place.
  2. General Irreducible Markov Chains and Non-Negative Operators by Nummelin. This is a thin and (very) concise book on general state space Markov chains. To be honest, it takes a lot of time to wrap your head around the notation. Not my favorite, but rigorous nonetheless.
  3. General State Space Markov Chains by Roberts and Rosenthal. This is not a book, but a survey paper on MCMC methods with detailed theory. This would be the perfect place to start for someone interested in MCMC theory. They also cite various books for readers to refer to.
  4. Handbook of Markov Chain Monte Carlo by Brooks, Gelman, Jones, and Meng. Not theoretical but definitely more recent (and in my view better) than other MCMC practice books.
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  • $\begingroup$ +1 for citing the Brooks et al. book, definitely the best for MCMC users. I'm curious to know why you didn't include Monte Carlo Statistical Methods by Robert and Casella - it's fairly theoretical and I think including it would make the answer more complete. Or do you have specific reasons for not recommending it? $\endgroup$ – DeltaIV Jul 26 '16 at 8:30
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    $\begingroup$ @DeltaIV I only included the Handbook as the one on the practice of MCMC and because I really like the book. I didn't include Robert + Casella because the OP in the comments had said that this does not have the answer to their question. The OP want's a book they can use to study the behavior of MCMC Markov chains, for which I personally don't think R+C is a good book. $\endgroup$ – Greenparker Jul 26 '16 at 9:39

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