- 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: 23 customer reviews
- Amazon Best Sellers Rank: #321,005 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.
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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.
Consider Section 6.5, 'Sum of non-negative random variables'. Out of the 4 (fundamental) statements made in there, 3 are 'proved' by merely stating that they are 'obvious','immediate','evident' etc
Now this is certainly the case for the author, but the rest of us are buying this book to understand the material, and this attitude borders on being disrespectful.
Again, one needs to buy this book for the examples inside, some of them famous , eg the 'ABRACADABRA' question, but to actually learn the material and have proper proofs - there are MUCH better books out there, covering the SAME material and with COMPLETE proofs: consider 'Probability Essentials' by Jacod for instance. Don't believe me ? Compare and contrast the proof of the Holder inequality for instance !
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.