Probability with Martingales (Cambridge Mathematical Textbooks) [Paperback]

David Williams (Author)

Book Description

ISBN-10: 0521406056 | ISBN-13: 978-0521406055 | Publication Date: February 22, 1991

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.

Editorial Reviews

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

Book Description

The author adopts the martingale theory as his main theme in this introduction to the modern theory of probability, which is, perhaps, at a practical level, one of the most useful mathematical theories ever devised.

Product Details

• Paperback: 251 pages
• Publisher: Cambridge University Press (February 22, 1991)
• Language: English
• ISBN-10: 0521406056
• ISBN-13: 978-0521406055
• Product Dimensions: 9 x 6 x 0.8 inches
• Shipping Weight: 12.6 ounces (View shipping rates and policies)
• Average Customer Review: 4.1 out of 5 stars  See all reviews (17 customer reviews)
• Amazon Best Sellers Rank: #80,422 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

28 of 28 people found the following review helpful:

5.0 out of 5 stars excellent probability text, January 23, 2008

By 

Michael R. Chernick "statman31147" (Holland PA) - See all my reviews
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This review is from: Probability with Martingales (Cambridge Mathematical Textbooks) (Paperback)

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.

47 of 51 people found the following review helpful:

5.0 out of 5 stars For the Probabilist who wants to travel light, February 9, 1998

By 

Giuseppe A. Paleologo "gappy" (Riverdale, NY United States) - See all my reviews
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This review is from: Probability with Martingales (Cambridge Mathematical Textbooks) (Paperback)

This textbook is an introduction to the measure-theoretic theory of probability. The style is unconventional. There is humor here, together with hints and suggestions for the "working probabilist". The first part of the book is rather conventional and introduces the concepts of probability spaces, events, expectation, independence of events. The second part introduces discrete-parameter martingales. Many results are given a "martingale proof". Usually, proofs are elegant and concise (at the cost of not being super-rigorous). For example, existence of conditional expectation is proved using ortogonal projection in L^2 (very nice!). Exercises are interesting and mixed with the text. There are no typos, and the cost of the book is reasonable. I would advise my grandma to buy this book (if she were interested in probability).

20 of 22 people found the following review helpful:

5.0 out of 5 stars eccentric, but wonderful, May 21, 2003

By 

R. D. Rivers (Palo Alto, CA USA) - See all my reviews
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This review is from: Probability with Martingales (Cambridge Mathematical Textbooks) (Paperback)

The reviewer who rated this a single star gives a decent imitation of Williams' prose style. What he doesn't mention is Williams' infectious enthusiasm for probability, the beautiful proofs, and the conciseness of this book. You should, of course, read Feller vol. 1 first, but this would be my next choice. I'd never really appreciated rigorous probability before reading this book. He shows that it's not all technicalities.