What books to read for MCMC theory? 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). 
 A: 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.


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*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.

*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.

*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.

*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.

