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General state space Markov chains and MCMC algorithms.

Roberts, Gareth O. and Rosenthal, Jeffrey S. (2004) General state space Markov chains and MCMC algorithms. Probability Surveys, 1. pp. 20-71. ISSN 1549-5787

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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.

Item Type: Article
Journal or Publication Title: Probability Surveys
Subjects: Q Science > QA Mathematics
Departments: Faculty of Science and Technology > Lancaster Environment Centre
ID Code: 9532
Deposited By: Mrs Yaling Zhang
Deposited On: 13 Jun 2008 11:15
Refereed?: Yes
Published?: Published
Last Modified: 26 Jul 2012 18:38
Identification Number:
URI: http://eprints.lancs.ac.uk/id/eprint/9532

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