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Probability with Martingales, by David Williams
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Probability theory is nowadays applied in a huge variety of fields including physics, engineering, biology, economics and the social sciences. This book is a modern, lively and rigorous account which has Doob's theory of martingales in discrete time as its main theme. It proves important results such as Kolmogorov's Strong Law of Large Numbers and the Three-Series Theorem by martingale techniques, and the Central Limit Theorem via the use of characteristic functions. A distinguishing feature is its determination to keep the probability flowing at a nice tempo. It achieves this by being selective rather than encyclopaedic, presenting only what is essential to understand the fundamentals; and it assumes certain key results from measure theory in the main text. These measure-theoretic results are proved in full in appendices, so that the book is completely self-contained. The book is written for students, not for researchers, and has evolved through several years of class testing. Exercises play a vital rôle. Interesting and challenging problems, some with hints, consolidate what has already been learnt, and provide motivation to discover more of the subject than can be covered in a single introduction.
- Sales Rank: #1058534 in eBooks
- Published on: 1991-02-14
- Released on: 1991-02-14
- Format: Kindle eBook
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
From the Back Cover
The book is written for students, not for researchers, and has evolved through several years of class testing. Exercises play a vital role. Interesting and challenging problems, some with hints, consolidate what has already been learnt, and provide motivation to discover more of the subject than can be covered in a single introduction.
About the Author
David Williams heads the four- and five-year-old division
David Williams heads the four- and five-year-old division of the summer soccer camps sponsored by Fort Lewis College, of the summer soccer camps sponsored by Fort Lewis College, in Durango, Colorado.
Scott Graham is an author, journal in Durango, Colorado.
Scott Graham is an author, journalist, father, and coach of young soccer players. ist, father, and coach of young soccer players.
Most helpful customer reviews
35 of 35 people found the following review helpful.
excellent probability text
By Michael R. Chernick
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.
51 of 56 people found the following review helpful.
For the Probabilist who wants to travel light
By Giuseppe A. Paleologo
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).
15 of 16 people found the following review helpful.
A nice treatment of Discreet Time Martingales
By Abdullah Alothman
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. Various versions of the Optional Sampling theorem - though not the most general, since uniform integrability and hence closure has not yet been defined -are proved.
Chapter 11: Only three pages. This motivates and gives a lovely proof of the Submartingale Upcrossing Theorem. The proof is so intuitive and simple, in marked contrast to that given in Billingsley. Various limit theorems - assuming L1 boundedness - are then proven, though none showing convergence in mean, for these the reader must wait till chapters 12 and 13.
Chapter 12: Defines the concept of L2 bounded martingales. Then digresses for 8 pages. This digression builds some machinery and uses it to prove both Kolmogorovs Three Series Theorem and the Strong Law of Large Numbers. The chapter ends by proving that every process X - in L1 - can be decomposed into:
X = X0 + M + A
Where A is Predicitable null at 0 and M is a Martingale null at 0.
In the case where X is a Submartingale A is shown to be increasing. This is the discreet version of the Doob Meyer decomposition, which says that every cadlag Submartingale is a Semimartingale.
Chapter 13: This introduces the concept of Uniform Integrability.
Chapter 14: Finally we are ready, armed with the machinery developed in 13, to prove convergence results for Uniformly Integrable Martingales. Convergence in L1, Levy's Upward / Downward theorems, Doobs SubMartingale Inequalities are all proven. Finally, a beautiful proof of the Radon Nikodym theorem is provided.
Chapter 15: Applications.
Part3: Three brief chapters dealing with, Characteristic Functions, Weak Convergence, the Central Limit Theorem, in that order. This part, consisting of 20 pages, would have been better left out. It is only the briefest of introductions to these areas, and therefore, given that this is a book on mathematics, should be left out. Instead, I refer the reader to Chapter 5, sections 25, 26 and 27, in Billingsley, for an excellent treatment of the above topics.
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