General state space Markov chains and MCMC algorithms

Details General state space Markov chains and MCMC algorithms General state space Markov chains and MCMC algorithms

2004-04-02

arxiv.org/abs/math/0404033v4

Probability Surveys 2004, Vol. 1, 20-71

Mathematics

Probability

Published at http://dx.doi.org/10.1214/154957804100000024 in the Probability Surveys (http://www.i-journals.org/ps/) by the In

Scientific paper

10.1214/154957804100000024

This paper surveys various results about Markov chains on general (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, 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 minorisation and drift conditions. Necessary and sufficient conditions for Central Limit Theorems (CLTs) 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.

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