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A First Course in Probability and Markov Chains

Giuseppe Modica, Laura Poggiolini

Wiley, 2013, xii + 334 pages, €62.00/£50.00/$85.00, hardcover

Table of Contents

  • 1.
    Combinatorics
  • 2.
    Probability measures
  • 3.
    Random variables
  • 4.
    Vector valued random variables
  • 5.
    Discrete time Markov chains
  • 6.
    An introduction to continuous time Markov chains
    • Power series

    • Measure and integration

    • Systems of linear ordinary differential equations

Readership: Mathematically well-prepared students who are taking a course in the theory of probability.

This is a mathematically demanding text that gives a theoretically rigorous account of the theory. Relevant measure theory, required for the second and later chapters, is summarised in Appendix B. The chapter on discrete time Markov chains forms an introduction to a simple type of Markov chain. It proceeds to introduce the theory of Markov chain Monte Carlo methods and gives a proof of the renewal theorem. The final chapter, on continuous time Markov chains, takes the Poisson process as its starting point. A limited number of examples that draw attention to practical applications are scattered through the text.

For a mathematically rigorous introduction to probability, this text seems a very reasonable choice. This reviewer would want to combine such a course with practice in practical computation involving probabilities. Corrigenda for the first printing are available from the website http://www.dma.unifi.it/$\sim$modica/libri/libri.html.