On the approximation of one Markov chain by another

Motivated by applications in Markov chain Monte Carlo, we discuss what it means for one Markov chain to be an approximation to another. Specifically included in that discussion are situations in which a Markov chain with continuous state space is approximated by one with finite state space. A simple sufficient condition for close approximation is derived, which indicates the existence of three distinct approximation regimes. Counterexamples are presented to show that these regimes are real and not artifacts of the proof technique. An application to the ``ball walk' of Lovász and Simonovits is provided as an illustrative example. Partially supported by EPSRC grant ``Sharper Analysis of Randomized Algorithms: a Computational Approach' and the IST Programme of the EU under contract IST-1999-14036 (RAND-APX). The work described here was partially carried out while the author was visiting the Isaac Newton Institute for Mathematical Sciences, Cambridge, UK. Postal address: School of Informatics, University of Edinburgh, The King's Buildings, Edinburgh EH9 3JZ, United Kingdom.