Content

1. Discrete-time- Markov chains
1.1 Definition and basic properties
1.2 Class structure
1.3 Hitting times and absorption probabilities
1.4 Strong Markov property
1.5 Recurrence and transience
1.6 Recurrence and transience of random walks
1.7 Invariant distributions
1.8 Convergence to equilibrium
1.9 Time reversal
1.10 Ergodic theorem
1.11 Appendix: recurrence relations
1.12 Appendix: asymptotics for n!

2. Continuous-time Markov chains I
2.1 Q-matrices and their exponentials
2.2 Continuous-time random processes
2.3 Some properties of the exponential distribution

2.4 Poisson processes 73
2.5 Birth processes 81
2.6 Jump chain and holding times 87
2.7 Explosion 90
2.8 Forward and backward equations 93
2.9 Non-minimal chains 103
2.10 Appendix: matrix exponentials 105
3. Continuous-time Markov chains II 108
3.1 Basic properties 108
3.2 Class structure 111
3.3 Hitting times and absorption probabilities 112
3.4 Recurrence and transience 114
3.5 Invariant distributions 117
3.6 Convergence to equilibrium 121
3.7 Time reversal 123
3.8 Ergodic theorem 125
4. Further theory 128
4.1 Martingales 128
4.2 Potential theory 134
4.3 Electrical networks 151
4.4 Brownian motion 159

5. Applications 170
5.1 Markov chains in biology 170
5.2 Queues and queueing networks 179
5.3 Markov chains in resource management 192
5.4 Markov decision processes 197
5.5 Markov chain Monte Carlo 206
6. Appendix: probability and measure 217
6.1 Countable sets and countable sums 217
6.2 Basic facts of measure theory 220
6.3 Probability spaces and expectation 222
6.4 Monotone convergence and Fubini's theorem 223
6.5 Stopping times and the strong Markov property 224
6.6 Uniqueness of probabilities and independence of a-algebras 228
Further reading 232
Index 234