- Series: Cambridge Mathematical Textbooks
- Paperback: 251 pages
- Publisher: Cambridge University Press; 1 edition (February 22, 1991)
- Language: English
- ISBN-10: 0521406056
- ISBN-13: 978-0521406055
- Product Dimensions: 6 x 0.7 x 9 inches
- Shipping Weight: 13.6 ounces (View shipping rates and policies)
- Average Customer Review: 22 customer reviews
- Amazon Best Sellers Rank: #170,727 in Books (See Top 100 in Books)
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Probability with Martingales (Cambridge Mathematical Textbooks) 1st Edition
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"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
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.
Top customer reviews
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.
I can see it working as a supplementary book but not at all as a primary textbook.
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.
On the other hand, even after learning the subject, one is always looking for something concise, consistently engaging, that gives a good view of the subject, allows you to make new connections, and gives you new ideas. Williams' book is all of that. It is not a book to have on a first exposure to the subject, maybe not for a second exposure either -- that will very much depend on what kind of student you are, and what you want to learn, and how you want to learn it. Only some very special students will go unaided through Williams' book on a first reading. But if you have some experience with the subject already (or with measure theory), and you want to broaden your horizons, then this book will allow you to do that. Williams' enthusiasm shines through every page, which is a plus. At this stage in my understanding of the subject, I actually appreciate that the book doesn't go into every detail, but shows more than enough to be a good guide. I didn't give it 5 stars because, to my taste, it should contain more exercises.
Having said all of that (about the book not being suitable for a first reading), I will take it all back, if you have the "correct" intructor teaching you the material: someone who will fill in some gaps when you need it, give you extra exercises, and in general give you that confidence that you need to feel that you are doing the right thing, and not just lost in the woods.
Most recent customer reviews
I used this book for self-study after struggling with Billingsley and Chung for months.Read more