General state space Markov chains and MCMC algorithms

This paper surveys various results about Markov chains on gen- eral (non-countable) state spaces. It begins with an introduction to Markov chain Monte Carlo (MCMC) algorithms, which provide the motivation and context for the theory which follows. Then, sucient conditions for geomet- ric and uniform ergodicity are presented, along with quantitative bounds on the rate of convergence to stationarity. Many of these results are proved using direct coupling constructions based on minorisation and drift con- ditions. Necessary and sucient conditions for Central Limit Theorems (CLTs) are also presented, in some cases proved via the Poisson Equa- tion or direct regeneration constructions. Finally, optimal scaling and weak convergence results for Metropolis-Hastings algorithms are discussed. None of the results presented is new, though many of the proofs are. We also describe some Open Problems.