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A note on acceptance rate criteria for CLTS for Metropolis–Hastings algorithms

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

G. O. Roberts
G. O. Roberts*
Affiliation:
Cambridge University
*
∗Postal address: Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, UK. Email address: g.o.roberts@lancaster.ac.uk.
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Abstract

This paper considers positive recurrent Markov chains where the probability of remaining in the current state is arbitrarily close to 1. Specifically, conditions are given which ensure the non-existence of central limit theorems for ergodic averages of functionals of the chain. The results are motivated by applications for Metropolis–Hastings algorithms which are constructed in terms of a rejection probability (where a rejection involves remaining at the current state). Two examples for commonly used algorithms are given, for the independence sampler and the Metropolis-adjusted Langevin algorithm. The examples are rather specialized, although, in both cases, the problems which arise are typical of problems commonly occurring for the particular algorithm being used.

Keywords

Hastings algorithmsMetropolis algorithmsMarkov chain Monte Carlocentral limit theorems

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Short Communications
Information
Journal of Applied Probability , Volume 36 , Issue 4 , December 1999 , pp. 1210 - 1217
Copyright
Copyright © Applied Probability Trust 1999 

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