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General state space Markov chains and MCMC algorithms. (English) Zbl 1189.60131

Summary: This paper surveys various results about Markov chains on general (non-countable) state spaces. It begins with an introduction to Markov chain Monte Carlo algorithms, which provide the motivation and context for the theory which follows. Then, sufficient conditions for geometric 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 minorization and drift conditions. Necessary and sufficient conditions for central limit theorems are also presented, in some cases proved via the Poisson equation 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.

MSC:

60J05 Discrete-time Markov processes on general state spaces
62F10 Point estimation
65C05 Monte Carlo methods
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
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