Probability. Theory and examples.

*(English)*Zbl 0709.60002
The Wadsworth & Brooks/Cole Statistics/Probability Series. Pacific Grove, CA: Wadsworth & Brooks/Cole Advanced Books & Software. ix, 453 p. (1991).

As the author states in the preface, the book is designed for a one-year graduate course in probability theory for students who are familiar with measure theory. To help those students who do not meet this prerequisite, the book contains a 35 pages appendix on the essentials of measure theory including some proofs and a number of exercises. The book consists of seven chapters: Laws of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic theorems and Brownian motion. Except for the last chapter, the author only treats the discrete-time case.

The author covers a suprisingly large amount of material including more modern developments, like Kingman’s subadditive ergodic theorem, a section on large deviations and a coupling proof of the renewal theorem as well as classics like infinitely divisible distributions. One special feature of the book is the large number of exercises and side remarks including applications, examples, counterexamples, improvements and historic remarks (even though the author states that “this is not a history book”) which certainly justify the title “Probability: Theory and Examples”.

The beginner in probability might be overwhelmed by the tight (though lively and pedagogically good) presentation of such an amount of material. A more advanced reader will certainly profit most from the book. There are not many really good modern textbooks on probability for this group of readers. This is certainly one of them; P. Billingsley, Probability and measure (1986; Zbl 0649.60001), is another one.

The author covers a suprisingly large amount of material including more modern developments, like Kingman’s subadditive ergodic theorem, a section on large deviations and a coupling proof of the renewal theorem as well as classics like infinitely divisible distributions. One special feature of the book is the large number of exercises and side remarks including applications, examples, counterexamples, improvements and historic remarks (even though the author states that “this is not a history book”) which certainly justify the title “Probability: Theory and Examples”.

The beginner in probability might be overwhelmed by the tight (though lively and pedagogically good) presentation of such an amount of material. A more advanced reader will certainly profit most from the book. There are not many really good modern textbooks on probability for this group of readers. This is certainly one of them; P. Billingsley, Probability and measure (1986; Zbl 0649.60001), is another one.

Reviewer: M.Scheutzow

##### MSC:

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60Fxx | Limit theorems in probability theory |

60G42 | Martingales with discrete parameter |

60E07 | Infinitely divisible distributions; stable distributions |

60G50 | Sums of independent random variables; random walks |