1 Introduction and results ..................................... 5
2 Finite state spaces ......................................... 11
2.1 Markov chains .......................................... 12
2.2 Error bounds ........................................... 17
2.3 Burn-in ................................................ 25
2.4 Examples ............................................... 27
2.5 Notes and remarks ...................................... 32
3 General state spaces ........................................ 35
3.1 Markov chains .......................................... 35
3.2 Error bounds ........................................... 49
3.3 Burn-in ................................................ 59
3.4 Examples ............................................... 63
3.5 Notes and remarks ...................................... 71
4 Applications ................................................ 76
4.1 Integration with respect to log-concave densities ...... 78
4.2 Integration over a convex body ......................... 82
4.3 Notes and remarks ...................................... 87
Appendix ....................................................... 88
A.l. Spectral theorem ....................................... 88
A.2. Interpolation theorem .................................. 89
A.3. Conductance and the Cheeger inequality ................. 89
References ..................................................... 90
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