Product Description

This is a masterly introduction to the modern and rigorous theory of probability. The author adopts the martingale theory as his main theme and moves at a lively pace through the subject's rigorous foundations. Measure theory is introduced and then immediately exploited by being applied to real probability theory. Classical results, such as Kolmogorov's Strong Law of Large Numbers and Three-Series Theorem are proved by martingale techniques. A proof of the Central Limit Theorem is also given. The author's style is entertaining and inimitable with pedagogy to the fore. Exercises play a vital role; there is a full quota of interesting and challenging problems, some with hints.

Product Details

• Amazon Sales Rank: #65732 in Books
• Published on: 1991-02-22
• Original language: English
• Number of items: 1
• Binding: Paperback
• 272 pages

Review
"Williams, who writes as though he were reading the reader's mind, does a brilliant job of leaving it all in. And well that he does, since the bridge from basic probability theory to measure theoretic probability can be difficult crossing. Indeed, so lively is the development from scratch of the needed measure theory, that students of real analysis, even those with no special interest in probability, should take note." D.V. Feldman, Choice

"...a nice textbook on measure-theoretic probability theory." Jia Gan Wang, Mathematical Reviews

Customer Reviews

excellent probability text
This is an excellently written text on probability theory that emphasizes the martingale approach. The treatment is softer than Neveu's "Discrete Parameter Martingales". Williams intends this book for third year undergraduates with good mathematical training as well as for graduate students.
It provides all the classic results including the Strong Law of Large Numbers and the Three-Series Theorem using martingale techniques for the proofs. It includes many exercises that the author encourages the reader to go through. The author recommends the texts of Billingsley, Chow and Teicher, Chung, Kingman and Taylor, Laha and Rohatgi and Neveu's 1965 probability theory book for a more thorough treatment of the theory.

Measure theory is at the heart of probability and Williams does not avoid it. Rather he embraces it and views probability as both a source of application for measure theory and a subject that enriches it. He covers the necessary measure theoretic groundwork.

However, advanced courses in probability that require measure theory are usually easier to grasp if the student has had a previous mathematics course in measure theory. In the United States, this usually doesn't occur until the fourth year and measure theory is mostly taken by undergraduate mathematics majors. Sometimes it is taken by first year graduate students concurrent with or prior to a course in advanced probability. For these reasons I would advise most instructors to consider it mainly for a graduate course in probability for math or statistics majors.

In the Preface, the author is quick to point out that probability is a subtle subject and honing one's intuition can be very important. He refers to Aldous' 1989 book as a source to help that process. I was disappointed that he didn't mention the two volumes on probability by Feller. Feller's books, particularly volume 2 with his treatment of the waiting time paradox, Benford's law and other puzzling problems in probability is a most stimulating source for appreciating the subtleties of probability, for honing one's intuition and for craving to learn more. It is a shame that Williams didn't mention it there. At some point Williams does refer to Feller's work but he only references volume 1.

Interesting book, but needs complementary material
I used this book as a companion through two courses, one in measure theory (including Lebesgue integration theory) and the second in advanced probability (ending just before stochastic integrations). The book is well written, but somewhat whimsical. Certain topics (such as integration) are covered somewhat superficially. For example, what Willams calls the 'standard machine' for integration (indicator functions to simple functions and then monotone convergence, etc.) is a very powerful idea but hastily described. Doob's theorems are well covered, and stopping times are also treated well. Unfortunately, many proofs are covered very hastily; call me stupid, but I need to refer to Billingsley or other text material in order to understand them.

But my favourite part of the book is the Brobdingnag sheep problem example. In most other books, examples of applications of martingale theory come from the hackneyed financial modelling world; this problem is not only an interesting riddle, but also lays the germs of stochastic optimal control, which to me (as an engineer) is probably of far more interest.

In summary, well worth the buy, but make sure that it is not the only thing you depend on. However, I found plenty of good supplementary material on the web, so that should not be a problem.

A Pedagogical Masterpiece
A Pedagogical Masterpiece
I used this book for self-study after struggling with Billingsley and Chung for months. Williams has a deep love for probability and it comes out rather beautifully in his writing. He gives the various topics just enough attention to maintain the brisk tempo necessary for a first course in measure theoretic probability and his proofs and exposition are on average much clearer than the aforementioned authors'. He has a knack of anticipating the average readers' stumbling blocks and states clearly which results are the fundamental ones and which aren't so important.

Read this book as a companion to Billingsley, Chung, or any other. Read it to lead up to Karatzas and Shreve, Rogers and Williams, or Protter. Or read it simply for inspiration.