Open Access
2004 General state space Markov chains and MCMC algorithms
Gareth O. Roberts, Jeffrey S. Rosenthal
Probab. Surveys 1: 20-71 (2004). DOI: 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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Gareth O. Roberts. Jeffrey S. Rosenthal. "General state space Markov chains and MCMC algorithms." Probab. Surveys 1 20 - 71, 2004.


Published: 2004
First available in Project Euclid: 8 November 2004

zbMATH: 1189.60131
MathSciNet: MR2095565
Digital Object Identifier: 10.1214/154957804100000024

Rights: Copyright © 2004 The Institute of Mathematical Statistics and the Bernoulli Society

Vol.1 • 2004
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