Probability with Martingales

by David Williams

This is a masterly introduction to the modern and rigorous theory of probability. The author adopts the martingale theory as his main theme and moves at a lively pace through the subject's rigorous foundations. Measure theory is introduced and then immediately exploited by being applied to real probability theory. Classical results, such as Kolmogorov's Strong Law of Large Numbers and Three-Series Theorem are proved by martingale techniques. A proof of the Central Limit Theorem is also given. The author's style is entertaining and inimitable with pedagogy to the fore. Exercises play a vital role; there is a full quota of interesting and challenging problems, some with hints.

Genres: Mathematics, Textbooks, Technical, Reference

251 pages, Paperback

First published January 1, 1991

About the author

Librarian Note: There is more than one author in the GoodReads database with this name. See this thread for more information.

David Williams is a Welsh mathematician who works in probability theory. He was educated at Gowerton Grammar School, winning a mathematics scholarship to Jesus College, Oxford, and went on to obtain a DPhil. He held posts at Stanford (1962–63), Durham, Cambridge (1966–69) and University College of Swansea (1969–85), where he was promoted to a personal chair in 1972. In 1985 he was elected to the Professorship of Mathematical Statistics, University of Cambridge, where he remained until 1992, serving as Director of the Statistical Laboratory between 1987 and 1991. Following this, he held the Chair of Mathematical Sciences jointly with the Mathematics and Statistics Groups at the University of Bath; he returned to Swansea in 1999, where he currently holds a Research Professorship.

Williams's research interests encompass Brownian motion, diffusions, Markov processes, martingales and Wiener–Hopf theory. Recognition for his work includes being elected Fellow of the Royal Society in 1984, where he was cited for his achievements on the construction problem for Markov chains and on path decompositions for Brownian motion, and being awarded the London Mathematical Society's Pólya Prize in 1994.

Reviews

[Dimitri · Rating 5 out of 5]

There is a large gap between classical and modern (measure theoretic) probability theory in that the later seems so much harder. However, without measure theory probability theory lacks a solid theoritical basis and leads to unsurmantouble problems in tryingto define stochastic processes.
I found this to be the best instructional book for those who want to transition from classical to the measure theoretic probability. I am not sure it is the best introductory book for measure theory and Lebesgue integration although the author offers a complete exposition in the first chapters. It is however an excellent exposition of how to apply these results in probability theory. By reading the book and doing the exercises the reader can gain intuition and start formulating probabilistic problems in this framework.

[Qubitng · Rating 2 out of 5]

Imagine watching the movie Pulp Fiction for the first time but without any sound. By the end of the movie, you might think you know what went on and you might even be right; but you would likely be wrong in all the details. Reading this book to learn measure theory or probability or martingales is just like watching Pulp Fiction without sound. If you know the details and loved it, you've probably seen the movie already with sound. If you didn't see it already with sound, you don't really know the details as well as you might think.

I can see it working as a supplementary book but not at all as a primary textbook.

[Shehryar Saroya · Rating 1 out of 5]

I have a lot of mixed feelings about this. This book is perhaps the densest mathematics book I've read so far. In no way did it hook me; I had to essentially force myself to read it because of my University course. It is essentially a set of barely tidied up lecture notes.

However, the issue is that it covers an area that is rarely taught at the undergraduate level in the US - so there is a distinct lack of approachable material in the area. Compared to its competition in the area, it's good; especially when supplemented with Rosenthal's: First Look at Rigorous Probability Theory.

[Evan · Rating 3 out of 5]

one of my favorite math texts, but it's definitely that. not a lot of "practical" examples - if by practical we mean computational - but written in an almost endearing manner totally uncustomary for a math text. not a reference, but excellent presentation for learning.

[Fausto Saleri · Rating 1 out of 5]

Inefficient, almost useless for any student. Notations are confuse, theorems are basically left to the reader; the choice behind the appendix is simply wrong, because you need to read it to go through the chapter, why separate things that have not to be separated?
It dares even to laugh at you with ridicolous spoilers, exactly like you would like to drop the read anytime. I never got to see something so childish and unprofessional in a math textbook.

[Joe Cole · Rating 5 out of 5]

The book is well written, but somewhat whimsical. Certain topics (such as integration) are covered somewhat superficially. 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.

[Mtaboga · Rating 5 out of 5]

Simple and concise. Essential reading for anyone interested in measure-theoretic probability.

[Risto Saarelma]

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