## Probability Surveys

### General state space Markov chains and MCMC algorithms

#### Abstract

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.

#### Article information

Source
Probab. Surveys, Volume 1 (2004), 20-71.

Dates
First available in Project Euclid: 8 November 2004

https://projecteuclid.org/euclid.ps/1099928648

Digital Object Identifier
doi:10.1214/154957804100000024

Mathematical Reviews number (MathSciNet)
MR2095565

Zentralblatt MATH identifier
1189.60131

#### Citation

Roberts, Gareth O.; Rosenthal, Jeffrey S. General state space Markov chains and MCMC algorithms. Probab. Surveys 1 (2004), 20--71. doi:10.1214/154957804100000024. https://projecteuclid.org/euclid.ps/1099928648

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