You are previewing Markov Chains.
O'Reilly logo
Markov Chains

Book Description

Markov chains are central to the understanding of random processes. This is not only because they pervade the applications of random processes, but also because one can calculate explicitly many quantities of interest. This textbook, aimed at advanced undergraduate or MSc students with some background in basic probability theory, focuses on Markov chains and quickly develops a coherent and rigorous theory whilst showing also how actually to apply it. Both discrete-time and continuous-time chains are studied. A distinguishing feature is an introduction to more advanced topics such as martingales and potentials in the established context of Markov chains. There are applications to simulation, economics, optimal control, genetics, queues and many other topics, and exercises and examples drawn both from theory and practice. It will therefore be an ideal text either for elementary courses on random processes or those that are more oriented towards applications.

Table of Contents

  1. Cover
  2. Title
  3. Copyright
  4. Dedication
  5. Preface
  6. Introduction
  7. 1. Discrete-time Markov chains
    1. 1.1 Definition and basic properties
    2. 1.2 Class structure
    3. 1.3 Hitting times and absorption probabilities
    4. 1.4 Strong Markov property
    5. 1.5 Recurrence and transience
    6. 1.6 Recurrence and transience of random walks
    7. 1.7 Invariant distributions
    8. 1.8 Convergence to equilibrium
    9. 1.9 Time reversal
    10. 1.10 Ergodic theorem
    11. 1.11 Appendix: recurrence relations
    12. 1.12 Appendix: asymptotics for n!
  8. 2. Continuous-time Markov chains I
    1. 2.1 Q-matrices and their exponentials
    2. 2.2 Continuous-time random processes
    3. 2.3 Some properties of the exponential distribution
    4. 2.4 Poisson processes
    5. 2.5 Birth processes
    6. 2.6 Jump chain and holding times
    7. 2.7 Explosion
    8. 2.8 Forward and backward equations
    9. 2.9 Non-minimal chains
    10. 2.10 Appendix: matrix exponentials
  9. 3. Continuous-time Markov chains II
    1. 3.1 Basic properties
    2. 3.2 Class structure
    3. 3.3 Hitting times and absorption probabilities
    4. 3.4 Recurrence and transience
    5. 3.5 Invariant distributions
    6. 3.6 Convergence to equilibrium
    7. 3.7 Time reversal
    8. 3.8 Ergodic theorem
  10. 4. Further theory
    1. 4.1 Martingales
    2. 4.2 Potential theory
    3. 4.3 Electrical networks
    4. 4.4 Brownian motion
  11. 5. Applications
    1. 5.1 Markov chains in biology
    2. 5.2 Queues and queueing networks
    3. 5.3 Markov chains in resource management
    4. 5.4 Markov decision processes
    5. 5.5 Markov chain Monte Carlo
  12. 6. Appendix: probability and measure
    1. 6.1 Countable sets and countable sums
    2. 6.2 Basic facts of measure theory
    3. 6.3 Probability spaces and expectation
    4. 6.4 Monotone convergence and Fubini’s theorem
    5. 6.5 Stopping times and the strong Markov property
    6. 6.6 Uniqueness of probabilities and independence of σ-algebras
  13. Further reading
  14. Index