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Probability with Martingales (Cambridge Mathematical Textbooks) Paperback – February 22, 1991

ISBN-13: 978-0521406055 ISBN-10: 0521406056

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Product Details

  • Series: Cambridge Mathematical Textbooks
  • Paperback: 251 pages
  • Publisher: Cambridge University Press (February 22, 1991)
  • Language: English
  • ISBN-10: 0521406056
  • ISBN-13: 978-0521406055
  • Product Dimensions: 8.9 x 6 x 0.5 inches
  • Shipping Weight: 13.6 ounces (View shipping rates and policies)
  • Average Customer Review: 4.1 out of 5 stars  See all reviews (19 customer reviews)
  • Amazon Best Sellers Rank: #95,892 in Books (See Top 100 in Books)

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.

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Customer Reviews

4.1 out of 5 stars
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I'd never really appreciated rigorous probability before reading this book.
Doug R
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.
Michael R. Chernick
A more detailed alternative that may serve as an excellent companion to this book is Probability and Measure by Ash.
Machine Learning Researcher

Most Helpful Customer Reviews

31 of 31 people found the following review helpful By Michael R. Chernick on January 23, 2008
Format: 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.
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49 of 54 people found the following review helpful By Giuseppe A. Paleologo on February 9, 1998
Format: 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).
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21 of 24 people found the following review helpful By Doug R on May 21, 2003
Format: 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.
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10 of 11 people found the following review helpful By Abdullah Alothman on January 24, 2006
Format: Paperback
Please Note:I gave it 5 stars because of Chapters 9 - 14.

This book consists of three parts. I will review each in turn:

Part 1: The First 8 Chapters, these covers basic measure theory:

The coverage here is streamlined and the pace is fast. I learnt measure theory as an undergraduate in the mid 80's using the text Measure Theory, also the first 8 chapters, by Halmos. The treatment here, if one is to take appendices 1-3 seriously, is almost at the level of Halmos, but the style, which is geared towards the probabilist, a lot more enjoyable. My only complaint, treatment of product measures and Fubini's theorem in section 8. One would do well to supplement this with the relevant section from Bilingsley's Probability and Measure.

Part2(The Core): Six Chapters on conditional expectation and discreet time martingale theory, one on applications:

The real beauty of this book is - modulu the chapter on applications, which like part three should have been left out - is in chapter 9 -14.

Chapter 9: Is a very nice treatment of conditional expectation. Its existence is proved using basic Hilbert Space Theory rather than the traditional Radon Nikodym - which the author does not develop in part 1-approach. Basic rules for its manipulation are then listed and proven. Armed with this, and the results from part 8 the reader is now finally ready to study martingale theory which is the subject of the next 5 Chapters.

Chapter 10: Is concerned with definitions. Martingales, Submartingales, and Super martingales - collectively called Smartingales, Chung's terminology - are defined. Optional times are defined. A very simple proof showing that Stopped Smartingales are Smartinglaes is given.
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5 of 5 people found the following review helpful By a reader on October 28, 2009
Format: Paperback
I have taught probability to undergraduates (in the US) from Sheldon Ross' A First Course in Probability, which I recommend to any student. I have also read Feller (superb) and Billingsley (superb) for myself, so you may imagine that I am not new to these topics. Learning from Ross is like learning calculus, learning from Billingsley is like learning mathematical analysis. One must progress from one to the other.

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.

Enjoy!
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