##
**Probability and random processes.
2nd ed.**
*(English)*
Zbl 0759.60001

Oxford: Oxford University Press,. xii, 541 p. (1992).

[For the first edition (1982) see Zbl 0493.60001.]

This textbook starts from the elements of probability, such as events and random variables, and moves on to more advanced topics such as discrete Markov chains in discrete time, discrete Markov processes in continuous time, stationary processes, spectral representation, the ergodic theorem, renewal theory and martingales. Also a number of special processes like birth-death processes, branching processes, queues and diffusion are treated in some detail.

The authors describe the book as undergraduate (and a reference for graduates), but presumably it is more at the graduate level. In fact, it is remarkable how many of the main basic topics of probability theory are given a rather complete introduction. Conditional expectations given \(\sigma\)-fields are used, but otherwise, the level of measure theory is very moderate. There is a great number of exercises and problems to each chapter. This number has been expanded in the second edition, and as a further novelty, a solution manual accompanies the book. Also special processes and martingales are given more attention compared to the first edition, and some modern ideas have entered, e.g. coupling as a tool for proving limit theorems for Markov chains.

The reviewer greatly enjoyed going through the book, not least due to its diversity, its up-to-date flavour and its many interesting and amusing examples. It is highly recommended as one of the nicest texts at its level.

This textbook starts from the elements of probability, such as events and random variables, and moves on to more advanced topics such as discrete Markov chains in discrete time, discrete Markov processes in continuous time, stationary processes, spectral representation, the ergodic theorem, renewal theory and martingales. Also a number of special processes like birth-death processes, branching processes, queues and diffusion are treated in some detail.

The authors describe the book as undergraduate (and a reference for graduates), but presumably it is more at the graduate level. In fact, it is remarkable how many of the main basic topics of probability theory are given a rather complete introduction. Conditional expectations given \(\sigma\)-fields are used, but otherwise, the level of measure theory is very moderate. There is a great number of exercises and problems to each chapter. This number has been expanded in the second edition, and as a further novelty, a solution manual accompanies the book. Also special processes and martingales are given more attention compared to the first edition, and some modern ideas have entered, e.g. coupling as a tool for proving limit theorems for Markov chains.

The reviewer greatly enjoyed going through the book, not least due to its diversity, its up-to-date flavour and its many interesting and amusing examples. It is highly recommended as one of the nicest texts at its level.

Reviewer: S.Asmussen (Aalborg)

### MSC:

60-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory |

60Gxx | Stochastic processes |

60Jxx | Markov processes |

60Kxx | Special processes |