Customer Reviews

Probability with Martingales (Cambridge Mathematical Textbooks)

5 star:		(10)
4 star:		(3)
3 star:		(2)
2 star:		(0)
1 star:		(2)

 

 

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.

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).

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.

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.

As the previous reviewers mentioned, this book is a concise and clear introduction to measure based probability. In contrast to some other reviewers, I like it a lot. The proofs ARE clear, the appendices ARE well placed and the order of presentation MAKES a lot of sense. One caveat is in order: since this book is EXTREMELY concise, one may have to think about the proofs and definitions for a while to digest it. The author does not chew up the material for you.

One more thing: This book is not just a rehashing of other textbooks on the topic. The order of presentation and proofs in this book is original (as far as I know) and if you find that you like it you are not likely to find it elsewhere.

A more detailed alternative that may serve as an excellent companion to this book is Probability and Measure by Ash.

Combine the book's originality of presentation with its clarity and its low price and the result is a great buy.

An excellent introduction to probability and integration.Assuming a minimal background in elementary probability andundergraduate real analysis, this book uses humor, interesting examples, and fun exercises (solvable, though usually non-trivial) to introduce modern probability theory. Unlike so many math books where you need an instructor to highlight what's important, this book makes a point of letting the reader know where to concentrate his/her time by assigning star ratings to the theorems.

Start reading this book, and soon you'll discover that you can learn more than you ever thought you could. The usual "Williams" style, but for the layman [almost]. Humor (humour) makes the book friendly. The section on a discrete-time version of the Black-Scholes formula can be improved... Could the author perhaps tell the story when the interest rate takes more than 2 values? In short, here is a book to cure those who think that (Omega, F, P) is redundant abstract nonsense. I also learned what "Mabinogion" stands for, but have not yet explained the seemingly "greek" sounding etymology. Maybe my Welsh needs to be improved :-) Takis Konstantopoulos

A Pedagogical Masterpiece
I used this book for self-study after struggling with Billingsley and Chung for months. Williams has a deep love for probability and it comes out rather beautifully in his writing. He gives the various topics just enough attention to maintain the brisk tempo necessary for a first course in measure theoretic probability and his proofs and exposition are on average much clearer than the aforementioned authors'. He has a knack of anticipating the average readers' stumbling blocks and states clearly which results are the fundamental ones and which aren't so important.

Read this book as a companion to Billingsley, Chung, or any other. Read it to lead up to Karatzas and Shreve, Rogers and Williams, or Protter. Or read it simply for inspiration.

It starts with very basic concepts of probability theory and covers a lot of topics in a very brief and up to the poit way. It does not go through much details. Beginners need to spend a lot of time on each page to understand what is going on.

I think this is the best introduction to modern probability and the theory of martingales. There is no unnecessary details, proofs are clear, exercises are interesting and challenging...

After reading this book, one may read "Diffusions, markov processes and martingales"